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An Introduction to Nuclear Physics
This clear and concise introduction to nuclear physics provides an excellent basis for a `core' undergraduate course in this area.
The book opens by setting nuclear physics in the context of elementary particle physics and then shows how simple models can provide an
understanding of the properties of nuclei, both in their ground states and
excited states, and also of the nature of nuclear reactions. The book
includes chapters on nuclear ®ssion, its application in nuclear power
reactors, and the role of nuclear physics in energy production and nucleosynthesis in stars.
This new edition contains several additional topics: muon-catalysed
fusion, the nuclear and neutrino physics of supernovae, neutrino mass
and neutrino oscillations, and the biological effects of radiation.
A knowledge of basic quantum mechanics and special relativity is
assumed. Appendices deal with other more specialised topics. Each chapter ends with a set of problems for which outline solutions are provided.
NOEL COTTINGHAM and DEREK GREENWOOD are theoreticians working in
the H. H. Wills Physics Laboratory at the University of Bristol. Noel
Cottingham is also a visiting professor at the Universite Pierre et Marie
Curie, Paris.
They have also collaborated in an undergraduate text, Electricity and
Magnetism, and a graduate text, An Introduction to the Standard Model of
Particle Physics. Both books are published by the Cambridge University
Press.
An Introduction to
Nuclear Physics
Second edition
W. N. COTTINGHAM
University of Bristol
D. A. GREENWOOD
University of Bristol
         
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
  
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
© Cambridge University Press 1986, 2004
First published in printed format 2001
ISBN 0-511-04046-6 eBook (netLibrary)
ISBN 0-521-65149-2 hardback
ISBN 0-521-65733-4 paperback
First published 1986
Reprinted 1987 (with corrections and additions), 1988, 1990, 1992, 1998
Second edition 2001
Contents
1
1.1
1.2
1.3
1.4
2
2.1
2.2
2.3
2.4
2.5
2.6
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Preface to the second edition
Preface to the ®rst edition
Constants of nature, conversion factors and notation
Glossary of some important symbols
Prologue
Fermions and bosons
The particle physicist's picture of nature
Conservation laws and symmetries: parity
Units
Problems
Leptons and the electromagnetic and weak interactions
The electromagnetic interaction
The weak interaction
Mean life and half life
Leptons
The instability of the heavy leptons: muon decay
Parity violation in muon decay
Problems
Nucleons and the strong interaction
Properties of the proton and the neutron
The quark model of nucleons
The nucleon±nucleon interaction: the phenomenological
description
Mesons and the nucleon±nucleon interaction
The weak interaction: ÿ-decay
More quarks
The Standard Model of particle physics
Problems
ix
x
xii
xiii
1
2
2
3
4
5
7
7
9
12
13
15
16
17
19
19
21
22
26
28
29
31
31
v
vi
Contents
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
6
6.1
6.2
6.3
7
7.1
7.2
7.3
7.4
7.5
8
8.1
8.2
8.3
8.4
9
9.1
Nuclear sizes and nuclear masses
Electron scattering by the nuclear charge distribution
Muon interactions
The distribution of nuclear matter in nuclei
The masses and binding energies of nuclei in their ground
states
The semi-empirical mass formula
The þ-stability valley
The masses of the þ-stable nuclei
The energetics of -decay and ®ssion
Nuclear binding and the nucleon±nucleon potential
Problems
Ground-state properties of nuclei: the shell model
Nuclear potential wells
Estimates of nucleon energies
Energy shells and angular momentum
Magic numbers
The magnetic dipole moment of the nucleus
Calculation of the magnetic dipole moment
The electric quadrupole moment of the nucleus
Problems
Alpha decay and spontaneous ®ssion
Energy release in -decay
The theory of -decay
Spontaneous ®ssion
Problems
Excited states of nuclei
The experimental determination of excited states
Some general features of excited states
The decay of excited states: ÿ-decay and internal conversion
Partial decay rates and partial widths
Excited states arising from þ-decay
Problems
Nuclear reactions
The Breit±Wigner formula
Neutron reactions at low energies
Coulomb effects in nuclear reactions
Doppler broadening of resonance peaks
Problems
Power from nuclear ®ssion
Induced ®ssion
33
33
36
37
39
41
44
48
50
52
52
56
56
58
60
65
66
67
68
72
74
74
75
83
87
89
89
93
97
99
100
101
103
103
107
109
111
113
115
115
Contents
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10
10.1
10.2
10.3
10.4
10.5
10.6
10.7
11
11.1
11.2
11.3
11.4
11.5
12
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
13
13.1
13.2
13.3
Neutron cross-sections for 235U and 238U
The ®ssion process
The chain reaction
Nuclear ®ssion reactors
Reactor control and delayed neutrons
Production and use of plutonium
Radioactive waste
The future of nuclear power
Problems
Nuclear fusion
The Sun
Cross-sections for hydrogen burning
Nuclear reaction rates in a plasma
Other solar reactions
Solar neutrinos
Fusion reactors
Muon-catalysed fusion
Problems
Nucleosynthesis in stars
Stellar evolution
From helium to silicon
Silicon burning
Supernovae
Nucleosynthesis of heavy elements
Problems
Beta decay and gamma decay
What must a theory of þ-decay explain?
The Fermi theory of þ-decay
Electron and positron energy spectra
Electron capture
The Fermi and Gamow±Teller interactions
The constants Vud and gA
Electron polarisation
Theory of ÿ-decay
Internal conversion
Problems
Neutrinos
Neutrino cross-sections
The mass of the electron neutrino
Neutrino mixing and neutrino oscillations
vii
116
118
119
121
122
124
125
126
127
130
130
132
135
139
140
143
146
148
151
151
155
156
157
160
161
163
163
166
168
171
173
177
178
179
184
185
186
186
188
189
viii
13.4
13.5
14
14.1
14.2
14.3
14.4
15
15.1
15.2
15.3
15.4
15.5
15.6
A.1
A.2
A.3
A.4
C.1
C.2
C.3
C.4
D.1
D.2
Contents
Solar neutrinos
Atmospheric neutrinos
Problems
The passage of energetic particles through matter
Charged particles
Multiple scattering of charged particles
Energetic photons
The relative penetrating power of energetic particles
Problems
Radiation and life
Ionising radiation and biological damage
Becquerels (and curies)
Grays and sieverts (and rads and rems)
Natural levels of radiation
Man-made sources of radiation
Risk assessment
Problems
Appendix A: Cross-sections
Neutron and photon cross-sections
Differential cross-sections
Reaction rates
Charged particle cross-sections: Rutherford scattering
Appendix B: Density of states
Problems
Appendix C: Angular momentum
Orbital angular momentum
Intrinsic angular momentum
Addition of angular momenta
The deuteron
Problems
Appendix D: Unstable states and resonances
Time development of a quantum system
The formation of excited states in scattering: resonances
and the Breit±Wigner formula
Problems
Further reading
Answers to problems
Index
193
195
196
199
199
206
207
211
212
214
214
215
216
217
218
219
220
222
222
224
225
226
227
230
230
230
232
233
234
235
235
236
241
244
245
246
267
Preface to the second edition
The main structure of the ®rst edition has been retained, but we have
taken the opportunity in this second edition to update the text and clarify
an occasional obscurity. The text has in places been expanded, and also
additional topics have been added. The growing interest of physics students in astrophysics has encouraged us to extend our discussions of the
nuclear and neutrino physics of supernovae, and of solar neutrinos. There
is a new chapter devoted to neutrino masses and neutrino oscillations. In
other directions, a description of muon-catalysed fusion has been
included, and a chapter on radiation physics introduces an important
applied ®eld.
We should like to thank Dr John Andrews and Professor Denis
Henshaw for their useful comments on parts of the text, Mrs Victoria
Parry for her secretarial assistance, and Cambridge University Press for
their continuing support.
W. N. Cottingham
D. A. Greenwood
Bristol, March 2000
ix
Preface to the ®rst edition
In writing this text we were concerned to assert the continuing importance
of nuclear physics in an undergraduate physics course. We set the subject
in the context of current notions of particle physics. Our treatment of
these ideas, in Chapters 1 to 3, is descriptive, but it provides a unifying
foundation for the rest of the book. Chapter 12, on þ-decay, returns to
the basic theory. It also seems to us important that a core course should
include some account of the applications of nuclear physics in controlled
®ssion and fusion, and should exemplify the role of nuclear physics in
astrophysics. Three chapters are devoted to these subjects.
Experimental techniques are not described in detail. It is impossible in
a short text to do justice to the ingenuity of the experimental scientist,
from the early discoveries in radioactivity to the sophisticated experiments of today. However, experimental data are stressed throughout:
we hope that the interdependence of advances in experiment and theory
is apparent to the reader.
We have by and large restricted the discussion of processes involving
nuclear excitation and nuclear reactions to energies less than about
10 MeV. Even with this restriction there is such a richness and diversity
of phenomena that it can be dif®cult for a beginner to grasp the underlying principles. We have therefore placed great emphasis on a few simple
theoretical models that provide a successful description and understanding of the properties of nuclei at low energies. The way in which simple
models can elucidate the properties of a complex system is one of the
surprises of the subject, and part of its general educational value.
We have tried to keep the mathematics as simple as possible. We
assume a knowledge of the basic formulae of special relativity, and
x
Preface to the first edition
xi
some basic quantum mechanics: wave-equations, energy levels and the
quantisation of angular momentum. A few topics which may not be
covered in elementary courses in quantum mechanics are treated in
appendices. We consider the technicalities of angular momentum algebra,
phase shift analysis and isotopic spin to be inappropriate to a ®rst course
in nuclear physics. Equations are written to be valid in SI units; results are
usually expressed in MeV and fm. Each chapter ends with a set of problems intended to amplify and extend the text; some refer to further
applications of nuclear physics. We have covered the bulk of the material
in this book in 35 lectures of the core undergraduate curriculum at
Bristol; these are given in the second and third years of the honours
physics course.
We thank colleagues and students who read drafts of the text and
drew our attention to errors and obscurities, which we have tried to
eliminate. We are grateful to Margaret James and Mrs Lilian Murphy
for their work on the typescript.
There is a less obvious debt: to the sometime Department of
Mathematical Physics of the University of Birmingham where, under
Professor Peierls, we ®rst learned about physics.
W. N. Cottingham
D. A. Greenwood
Bristol, August 1985
Constants of nature, conversion factors
and notation
Velocity of light
Planck's constant
Proton charge
Boltzmann's constant
c
» ˆ h=2
e
kB
Gravitational constant
Fermi coupling
constant
Electron mass
G
GF
Proton mass
Neutron mass
Atomic mass unit
Bohr magneton
Nuclear magneton
Bohr radius
Fine-structure
constant
2:997 92 108 m sÿ1
1:054 57 10ÿ34 J s
1:602 18 10ÿ19 C
1:380 7 10ÿ23 J Kÿ1
ˆ 8:617 10ÿ5 eV Kÿ1
6:67 10ÿ11 m3 kgÿ1 sÿ2
1:166 10ÿ11 »c†3 MeVÿ2
9:109 4 10ÿ31 kg
ˆ 0:511 00 MeV=c2
mp
1:007 276 amu
ˆ 938:27 MeV=c2
mn
1:008 66 amu
ˆ 939:57 MeV=c2
ÿ27
12
(mass C atom)/12 1:660 54 10
kg
ˆ 931:49 MeV=c2
B ˆ e »=2me
5:788 38 10ÿ11 MeV Tÿ1
N ˆ e »=2mp
3:152 45 10ÿ14 MeV Tÿ1
2
2
a0 ˆ 4"0 » =me e 0:529 177 10ÿ10 m
e2 =4"0 »c
1/137.036
me
»c ˆ 197:327 MeV fm; e2 =4"0 ˆ 1:439 96 MeV fm
1 MeV ˆ 1:602 18 10ÿ13 J
1 fm ˆ 10ÿ15 m; 1 barn ˆ 10ÿ28 m2 ˆ 102 fm2
(Source: Review of Particle Physics (1998), Eur. Phys. J. C3, 1±794.)
Notation
r, k, etc., denote vectors x; y; z†; kx ; ky ; kz †, and r ˆ jrj, k ˆ jkj,
d3 r ˆ dx dy dz; d3 k ˆ dkx dky dkz .
r2 ˆ
@2
@2
@2
1 @2
1
@
@
1
@2
‡ 2‡ 2ˆ
r‡ 2
,
sin ‡ 2 2
2
2
r @r
@ r sin @2
@x
@y
@z
r sin @
dþ ˆ sin d d denotes an in®nitesimal element of solid angle.
xii
Glossary of some important symbols
A
A r; t†
a
B Z; N†
B r; t†
b
E r; t†
E
F Z; Ee †
f Z; E0 †
G
Gw
g
gL ; gs
gA
G rs =rc †
J
j
jz
k
kF
L
l
nuclear mass number ˆ N ‡ Z†
electromagnetic vector potential
}4.1 nuclear surface width; }4.5 bulk binding coef®cient
binding energy of nucleus
magnetic ®eld
}4.5 surface tension coef®cient; }14.1 impact parameter
electric ®eld
energy; En , Ep neutron energy, proton energy; EnF , EpF
neutron, proton Fermi energy, measured from the bottom of the shell-model neutron potential well; EG }8.3
}12.3 Coulomb correction factor in þ-decay
}12.3 kinematic factor in total þ-decay rate
}6.2 exponent in the tunnelling formula
}12.2 weak interaction coupling constant ˆ GF Vud †
}8.1 statistical factor in Breit±Wigner formula
}5.6 orbital and intrinsic magnetic moment coef®cients
}12.5 axial coupling constant
}6.2 tunnelling integral
}C.3 total angular momentum operator
quantum number associated with J2
quantum number of Jz
wave vector
value of k ˆ jkj at the Fermi energy
}C.1 orbital angular momentum operator
quantum number associated with L2 ; Chapter 9,
Chapter 14 mean free path
xiii
xiv
Glossary of some important symbols
m
ms
m
N
n E†
N E†
p
Q
q
R
rs ; rc
Sn N; Z†
S E†
S0 E†; Sc E†
s
s
T
T1=2
tnuc
tp
U
ul r†
V
Vud
v
Z
ÿ; ÿi
ÿ
"0
"F
w
l
quantum number of Lz ; reduced mass
quantum number of sz
mass of -particle; ma , mnuc mass of atom, nucleus
number of neutrons in nucleus
density of states
integrated density of states
momentum
}5.7 nuclear electric quadrupole moment; }6.1 kinetic
energy release in nuclear reaction
}9.4 ®ssion probability
}4.3 nuclear radius; }12.3 reaction rate
}6.2 potential barrier parameters
}5.2 neutron separation energy
}8.3 parameter of nuclear reaction cross-section for
energies below the Coulomb barrier
}12.3 electron (positron) energy spectrum without and
with Coulomb correction
}C.2 intrinsic angular momentum operator
quantum number associated with s; }4.5 symmetry
energy coef®cient
kinetic energy
decay half life
}5.2 nuclear time scale
}9.4 prompt neutron life
potential energy; U mean proton±neutron potential
energy difference in nucleus
radial wave-function
normalisation volume; }3.3 V r† nucleon±nucleon
potential
}12.5 element of Kobayashi±Maskawa matrix
velocity
atomic number (number of protons in nucleus)
width, partial width, of an excited state
1
}14.1 relativistic factor 1 ÿ v2 =c2 †ÿ2
}4.4 coef®cient of pairing energy
permittivity of free space
}11.1 Fermi energy of electron gas
}13.3 neutrino mixing angle
}13.1 Weinberg angle
}5.5 magnetic dipole operator
Glossary of some important symbols
n ; p
0
,d
ch
0
nuc ; n ; p
r
ý r†
ým
þS0 ; þT
!
xv
neutron, proton magnetic moment
}5.5 magnetic dipole moment; }11.1 stellar mass per
electron; }14.3 photon linear attenuation coef®cient
permeability of free space
}9.3 mean number of prompt neutrons, delayed
neutrons, per ®ssion
}2.1 electric charge density; }14.1 mass density
}4.1 electric charge density in units of e
}4.3 nucleon number density in nuclear matter
number density of nuclei, neutrons, protons
}C.2 Pauli spin matrices
cross-section; tot , e , f total, elastic, ®ssion crosssection
mean life; E1 , M1 electric, magnetic, dipole transition
mean life; }7.4 i †ÿ1 partial decay rate
}3.4 meson ®eld
electromagnetic scalar potential
single particle wave-function
}D.1 general wave-function
}3.3 angular terms in the nucleon±nucleon potential
angular frequency
1
Prologue
More than 100 elements are now known to exist, distinguished from each
other by the electric charge Ze on the atomic nucleus. This charge is
balanced by the charge carried by the Z electrons which together with
the nucleus make up the neutral atom. The elements are also distinguished by their mass, more than 99% of which resides in the nucleus.
Are there other distinguishing properties of nuclei? Have the nuclei been
in existence since the beginning of time? Are there elements in the
Universe which do not exist on Earth? What physical principles underlie
the properties of nuclei? Why are their masses so closely correlated with
their electric charges? Why are some nuclei radioactive? Radioactivity is
used to man's bene®t in medicine. Nuclear ®ssion is exploited in power
generation. But man's use of nuclear physics has also posed the terrible
threat of nuclear weapons.
This book aims to set out the basic concepts which have been developed by nuclear physicists in their attempts to understand the nucleus.
Besides satisfying our appetite for knowledge, these concepts must be
understood if we are to make an informed judgment on the bene®ts
and problems of nuclear technology.
After the discovery of the neutron by Chadwick in 1932, it was
accepted that a nucleus of atomic number Z was made up of Z protons
and some number N of neutrons. The proton and neutron were then
thought to be elementary particles, although it is now clear that they
are not but rather are themselves structured entities. We shall also see
that in addition to neutrons and protons several other particles play an
important, if indirect, role in the physics of nuclei. In this and the following two chapters, to provide a background to our subsequent study of the
1
2
Prologue
nucleus, we shall describe the elementary particles of nature, and their
interactions, as they are at present understood.
1.1
Fermions and bosons
Elementary particles are classi®ed as either fermions or bosons. Fermions
are particles which satisfy the Pauli exclusion principle: if an assembly of
identical fermions is described in terms of single-particle wave-functions,
then no two fermions can have the same wave-function. For example,
electrons are fermions. This rule explains the shell structure of atoms and
hence underlies the whole of chemistry. Fermions are so called because
they obey the Fermi±Dirac statistics of statistical mechanics.
Bosons are particles which obey Bose±Einstein statistics, and are
characterised by the property that any number of particles may be
assigned the same single-particle wave-function. Thus, in the case of
bosons, coherent waves of macroscopic amplitude can be constructed,
and such waves may to a good approximation be described classically.
For example, photons are bosons and the corresponding classical ®eld is
the familiar electromagnetic ®eld E and B, which satis®es Maxwell's
equations.
At a more fundamental level, these properties are a consequence of
the possible symmetries of the wave-function of a system of identical
particles when the coordinates of any two particles are interchanged. In
the case of fermions, the wave-function changes sign; it is completely antisymmetric. In the case of bosons the wave-function is unchanged; it is
completely symmetric.
There is also an observed relation between the intrinisc angular
momentum, or spin, of a particle and its statistics. The intrinsic spin s
is quantised, with spin quantum number s (see Appendix C). For a fermion, s takes one of the values 12, 32, 52, . . .; for a boson, s takes one of the
values 0, 1, 2, . . . . A theoretical explanation of this relationship can be
given within the framework of relativistic quantum ®eld theory.
1.2
The particle physicist's picture of nature
Elementary particle physics describes the world in terms of elementary
fermions, interacting through ®elds of which they are sources. The particles associated with the interaction ®elds are bosons. To take the most
familiar example, an electron is an elementary fermion; it carries electric
charge ±e and this charge produces an electromagnetic ®eld E, B, which
1.3 Conservation laws and symmetries: parity
3
Table 1.1. Types of interaction ®eld
Interaction ®eld
Boson
Spin
Gravitational ®eld
Weak ®eld
Electromagnetic ®eld
Strong ®eld
`Gravitons' postulated
W+, Wÿ, Z particles
Photons
`Gluons' postulated
2
1
1
1
exerts forces on other charged particles. The electromagnetic ®eld, quantised according to the rules of quantum mechanics, corresponds to an
assembly of photons, which are bosons. Indeed, Bose±Einstein statistics
were ®rst applied to photons.
Four types of interaction ®eld may be distinguished in nature (see
Table 1.1). All of these interactions are relevant to nuclear physics,
though the gravitational ®eld becomes important only in densely aggregated matter, such as stars. Gravitational forces act on all particles and
are important for the physics on the large scale of macroscopic bodies. On
the small scale of most terrestrial atomic and nuclear physics, gravitational forces are insigni®cant and except in Chapter 10 and Chapter 11 we
shall ignore them.
Nature provides an even greater diversity of elementary fermions
than of bosons. It is convenient to divide the elementary fermions into
two classes: leptons, which are not sources of the strong ®elds and hence
do not participate in the strong interaction; and quarks, which take part
in all interactions.
The electron is an example of a lepton. Leptons and their interactions
are described in Chapter 2. Quarks are always con®ned in compound
systems which extend over distances of about 1 fm. The term hadron is
used generically for a quark system. The proton and neutron are hadrons,
as are mesons. The proton and neutron are the subject matter of
Chapter 3.
1.3
Conservation laws and symmetries: parity
The total energy of an isolated system is constant in time. So also are its
linear momentum and angular momentum. These conservation laws are
derivable from Newton's laws of motion and Maxwell's equations, or
from the laws of quantum mechanics, but they can also, at a deeper
4
Prologue
level, be regarded as consequences of `symmetries' of space and time.
Thus the law of conservation of linear momentum follows from the
homogeneity of space, the law of conservation of angular momentum
from the isotropy of space; it does not matter where we place the origin
of our coordinate axes, or in which direction they are oriented.
These conservation laws are as signi®cant in nuclear physics as elsewhere, but there is another symmetry and conservation law which is of
particular importance in quantum systems such as the nucleus: re¯ection
symmetry and parity. By re¯ection symmetry we mean re¯ection about
the origin, r ! r0 ˆ ÿr. A single-particle wave-function ÿ r† is said to
have parity ‡1 if it is even under re¯ection, i.e.
ÿ ÿr† ˆ ÿ r†;
and parity ÿ1 if it is odd under re¯ection, i.e.
ÿ ÿr† ˆ ÿÿ r†:
More generally, a many-particle wave-function has parity ‡1 if it is even
under re¯ection of all the particle coordinates, and parity ÿ1 if it is odd
under re¯ection.
Parity is an important concept because the laws of the electromagnetic and of the strong interaction are of exactly the same form if written
with respect to a re¯ected left-handed coordinate system 0x0 ; 0y0 ; 0z0 † as
they are in the standard right-handed system 0x; 0y; 0z† (Fig. 1.1). We
shall see in Chapter 2 that this is not true of the weak interaction.
Nevertheless, for many properties of atomic and nuclear systems the
weak interaction is unimportant and wave-functions for such systems
can be chosen to have a de®nite parity which does not change as the
wave-function evolves in time according to SchroÈdinger's equation.
1.4
Units
Every branch of physics tends to ®nd certain units particularly congenial.
In nuclear physics, the size of the nucleus makes 10ÿ15 m ˆ 1 fm (femtometre) convenient as a unit of length, usually called a fermi. However,
nuclear cross-sections, which have the dimensions of area, are measured
in barns; 1 b ˆ 10ÿ28 m2 ˆ 100 fm2 . Energies of interest are usually of the
order of MeV. Since mc2 has the dimensions of energy, it is convenient to
quote masses in units of MeV/c2 .
Problems
5
Fig. 1.1 The point P at r with coordinates x; y; z† has coordinates ÿx; ÿy; ÿz†
in the primed, re¯ected coordinate axes. 0x0 ; 0y0 ; 0z0 † make up a left-handed set of
axes. In the ®gure, the 0z axis is out of the plane of the page.
For order-of-magnitude calculations, the masses me and mp of the
electron and proton may be taken as
me 0:5 MeV=c2
mp 938 MeV=c2
and it is useful to remember that
»c 197 MeV fm;
e2 =4"0 1:44 MeV fm;
e2 =4"0 »c 1=137;
c 3 1023 fm sÿ1 :
The student will perhaps be surprised to ®nd how easily many expressions
in nuclear physics can be evaluated using these quantities.
Problems
1.1 Show that the ratio of the gravitational potential energy to the Coulomb
potential energy between two electrons is 2:4 10ÿ43 .
1.2(a) Show that in polar coordinates r; ; † the re¯ection
r ! r0 ˆ ÿr is equivalent to r ! r0 ˆ r
! 0 ˆ ÿ ; ! 0 ˆ ‡ .
6
Prologue
(b) What are the parities of the following electron states of the hydrogen
atom:
32
1
1
(i)
ÿ100 ˆ p
eÿr=a0 ,
a0
(ii)
32
1
1 r ÿr=2a0
ÿ210 ˆ p
e
cos ,
4 2† a0 a0
(iii)
32
1
1 r ÿr=2a0
e
sin eÿi ?
ÿ21ÿ1 ˆ p
8 a0 a0
(a0 ˆ 4"0 † »2 =me e2 is the Bohr radius.)
1.3(a) Show that the wavelength of a photon of energy 1 MeV is 1240 fm.
(b) The electrostatic self-energy of a uniformly charged sphere of total
charge e, radius R, is U ˆ 3=5†e2 = 4"0 R†. Show that if R ˆ 1 fm,
U 0:86 MeV.
2
Leptons and the electromagnetic and weak
interactions
2.1
The electromagnetic interaction
The electromagnetic ®eld is most conveniently described by a vector
potential A and a scalar potential . For simplicity, we consider only
the potential r; t†. Using Maxwell's equations, this may be chosen to
satisfy the wave-equation
r2 ÿ
1 @2 r; t†
ˆÿ
:
2
2
"0
c @t
2:1†
Here r; t† is the electric charge density due to the charged particles,
which in atomic and nuclear physics will usually be electrons and protons,
and c is the velocity of light.
In regions where ˆ 0, equation (2.1) has solutions in the form of
propagating waves; for example, the plane wave
r; t† ˆ constant† ei krÿ!t† :
2:2†
This satis®es
r2 ÿ
1 @2 ˆ0
c2 @t2
2:3†
provided
! 2 ˆ c 2 k2 :
2:4†
7
8
Leptons and the electromagnetic and weak interactions
The wave velocity is therefore c, as we should expect. In quantum theory,
unlike classical theory, the total energy and momentum of the wave are
quantised, and can only be integer multiples of the basic quantum of
energy and momentum given by the de Broglie relations:
E ˆ »!;
p ˆ »k:
2:5†
Such a quantum of radiation is called a photon. A macroscopic wave can
be considered to be an assembly of photons, and we can regard photons
as particles, each carrying energy E and momentum p.
Using (2.4) and (2.5), E and p are related by
E 2 ˆ p2 c2 :
2:6†
For a particle of mass m, the Einstein equation gives
E 2 ˆ p2 c2 ‡ m2 c4 :
We therefore infer that the photon is a particle having zero mass.
A second important type of solution of (2.1) exists when charged
particles are present. If these are moving slowly compared with the velocity of light, so that the term @2 = c2 @t2 † can be neglected, the solution is
approximately the Coulomb potential of the charge distribution. For a
particle with charge density 1 , we can take
r; t† 1
4"0
Z
1 r0 ; t† 3 0
d r:
jr ÿ r0 j
2:7†
Another charged particle with charge density 2 will have a potential
energy given by
Z
2 r; t† r; t†d3 r
Z
1
1 r0 ; t†2 r; t† 3 3 0
ˆ
d rd r :
4"0
jr ÿ r0 j
U12 ˆ
2:8†
Electric potential energy is basically responsible for the binding of
electrons in atoms and molecules. We shall see that, in nuclear physics, it
is responsible for the instability of heavy nuclei. If magnetic effects due to
the motion of the charges are included, equation (2.8) is modi®ed to
2.2 The weak interaction
U12 ˆ
1
4"0
Z
01 2 ‡ 1=c2 †j01 j2 3 3 0
d rd r ;
jr ÿ r0 j
9
2:9†
where j ˆ v is the current associated with the charge distribution which
has velocity v r†. Thus this magnetic contribution to the energy is of
relative order v2 =c2 .
The electromagnetic interaction also gives rise to the scattering of
charged particles. For example, for two electrons approaching each
other the interaction gives a mutual repulsion which leads to a transfer
of momentum between the particles. The process can be represented by a
diagram such as Fig. 2.1. In quantum electrodynamics, these diagrams,
invented by Feynman, have a precise technical interpretation in the theory. We shall use them only to help visualise the physics involved. The
scattering of the two electrons may be thought of as caused by the emission of a `virtual' photon by one electron and its absorption by the other
electron. In a virtual process the photon does not actually appear to an
observer, though it appears in the mathematical formalism that describes
the process.
2.2
The weak interaction
There are three weak interaction ®elds associated with the W+, Wÿ and Z
particles. Each one, like the electromagnetic ®eld, is described by a vector
and a scalar potential. However, the bosons associated with the weak
Fig. 2.1 The scattering of two electrons of momenta »k, »k0 by the exchange of a
virtual photon carrying momentum »q. Time runs from left to right in these
diagrams. (In principle, the exchange of a Z particle (}2.2) also contributes to
electron±electron scattering, but the very short range and weakness of the weak
interaction makes this contribution almost completely negligible: the electrons are
in any case kept apart by the Coulomb repulsion induced by the photon
exchange.)
10
Leptons and the electromagnetic and weak interactions
®elds all have mass, and the Wÿ and W+ bosons are electrically charged.
The Z boson is neutral, and most similar to the photon, but it has a mass
MZ ˆ 91:187 0:007† GeV=c2 100 proton masses;
which is very large by nuclear physics standards.
The interactions between leptons and the electromagnetic and weak
®elds were combined into a uni®ed `electro-weak' theory by Weinberg
and by Salam. The existence of the Z and W bosons was predicted by
the theory, and the theory together with experimental data from neutrino±nuclear scattering also suggested values for their masses. These
predictions were con®rmed by experiments at CERN in 1983.
The wave equation satis®ed by the scalar potential Z associated with
the Z boson is a generalisation of (2.1) and includes a term involving MZ :
"
#
1 @2
MZ c 2
r; t†
r ÿ 2 2ÿ
;
Z r; t† ˆ ÿ Z
»
"0
c @t
2
2:10†
where Z is the neutral weak-charge density. There is a close, but not
exact, analogy between weak-charge density and electric-charge density,
and particles carry weak charge somewhat as they carry electric charge. In
the case of a nucleus, Z will extend over the nuclear dimensions.
In free space where Z ˆ 0 there exist plane wave solutions of (2.10),
Z r; t† ˆ constant† ei krÿ!t† ;
but now to satisfy the wave equation we require
!2 ˆ c2 k2 ‡ c2 MZ c= »†2 ;
and with the de Broglie relations (2.5) for the ®eld quanta we obtain the
Einstein energy±momentum relation for the Z boson:
E 2 ˆ p2 c2 ‡ MZ2 c4 :
The static solution of (2.10) which corresponds to a point unit weak
charge at the origin is
Z r† ˆ
1 eÿr
;
4"0 r
writing ˆ
MZ c
:
»
2:11†
2.2 The weak interaction
11
At points away from the origin where r2 Z ÿ 2 Z ˆ 0, this satis®es
equation (2.10), as may be easily checked by substitution, using the formula r2 Z ˆ 1=r†d2 rZ †=dr2 . Close to the origin the solution (2.11)
behaves like the corresponding Coulomb potential 1= 4"0 r† of a unit
point electric charge, and hence has the correct point source behaviour.
The generalisation of (2.11) to a distribution of weak charge gives the
quasi-static solution (cf. (2.7))
Z r; t† 1
4"0
Z
0
Z r0 ; t† eÿjrÿr j 3 0
d r:
jr ÿ r0 j
2:12†
The exponential factor in the integral effectively vanishes for jr ÿ r0 j
greater than a few times ÿ1 ˆ »=MZ c and
»=MZ c 2 10ÿ3 fm:
This is a very small distance compared with the size of the nucleus. Hence
in the integral in (2.12) the factor Z is slowly varying over the range of
the exponential and may be taken outside the integral (which is then
elementary):
1
Z r; t† '
r; t†
4"0 Z
Z
0
eÿjrÿr j 3 0
d r
jr ÿ r0 j
Z 1 ÿR
1
e
4R2 dR
ˆ
r; t†
R
4"0 Z
0
1
» 2
Z r; t†:
ˆ
"0 M Z c
2:13†
The potential energy between two particles associated with the scalar ®eld
Z is, by analogy with (2.8),
Z
ˆ
U12
ˆ
Z
Z2 r; t†Z1 r; t†d3 r
2 Z
1
»
Z1 r; t†Z2 r; t†d3 r;
"0 M Z c
and there is also a contribution from the vector part of the ®eld, analogous to the magnetic contribution in (2.9), of the form
12
Leptons and the electromagnetic and weak interactions
Z
1
» 2
jZ1 r; t† jZ2 r; t†d3 r;
"0 c2 MZ c
where jZ is the weak-current density.
The physical consequences of these expressions are quite different
Z
from the physical consequences of the electromagnetic interaction. U12
is very much suppressed by the large mass factor in the denominator, and
it is this which largely accounts for the `weakness' of the weak interaction.
Also the interaction at low energies appears as a `contact interaction',
effectively having zero range.
The electrically charged W+ and Wÿ boson ®elds give rise to the
most important weak interactions, and in particular to þ-decay. They
obey equations similar to those of the Z ®eld, but the masses of the
associated particles are somewhat smaller;
MW‡ ˆ MWÿ ˆ 80:41 0:10† GeV=c2 :
2.3
Mean life and half life
Not all particles are stable: some, for example the W and Z bosons, have
only a transient existence. Suppose that an unstable particle exists at some
instant t ˆ 0; its mean life is the mean time it exists in isolation, before it
undergoes radioactive decay. If we denote by P t† the probability that the
particle survives for a time t, and make the assumption that the particle
has a constant probability 1= per unit time of decaying, then
P t ‡ dt† ˆ P t† 1 ÿ dt=†;
since 1 ÿ dt=† is the probability it survives the time interval dt. Hence
1 dP
1
ˆÿ ;
P dt
and integrating,
P t† ˆ P 0†eÿt= :
Since P 0† ˆ 1 we have
P t† ˆ eÿt= :
2:14†
2.4 Leptons
13
Equation (2.14) is the familiar exponential-decay law for unstable particles. It is well veri®ed experimentally.
The probability that the particle decays between times t, t ‡ dt is
clearly P t† dt=†, so that the mean life is
Z
1
0
Z
tP t† dt=† ˆ
1
0
teÿt= dt= ˆ :
The `half life' T12 is the time at which there is a 50% probability that
the particle has decayed, i.e.
ÿT1 =
P T12 † ˆ e
2
ˆ 12 :
Hence
T12 ˆ ln 2 ˆ 0:693:
In this book we have preferred to quote mean lives rather than half
lives. We refer to 1=† as the decay rate.
2.4
Leptons
Leptons are spin 12 fermions which interact through the electromagnetic
and weak interactions, but not through the strong interaction. The known
leptons are listed in Table 2.1.
The electrically charged leptons all have magnetic moments of magnitude e »=2 (mass) anti-aligned with their spins.
Of these charged leptons, only the familiar electron is stable.
Electrons are structureless particles that are described by the Dirac relativistic wave-equation. This equation explains the spin and magnetic
moment of the electron, and has the remarkable feature that it predicts
the existence of anti-particles: these are particles of the same mass and
spin, but of opposite charge and magnetic moment to the particle. The
anti-particle of the electron is called the positron. Positrons were identi®ed
experimentally by Anderson in 1932 soon after their theoretical prediction.
Since leptons do not interact with the strong interaction ®eld, electrons and positrons interact principally through the electromagnetic ®eld.
A positron will eventually annihilate with an electron, usually to produce
14
Leptons and the electromagnetic and weak interactions
Table 2.1. Known leptons
Electron eÿ
Electron neutrino e
Muon ÿ
Muon neutrino Tau ÿ
Tau neutrino Mass
(MeV/c2 †
Mean life
(s)
Charge
0.5110
< 15 10ÿ6
105.658
< 0:17
1777
< 18:2
1
1?
2:197 10ÿ6
1?
290 10ÿ15
1?
ÿe
0
ÿe
0
ÿe
0
two or three photons, so that all the lepton energy appears as electromagnetic radiation. We write these processes as
eÿ ‡ e‡ ! 2ÿ
eÿ ‡ e‡ ! 3ÿ:
Annihilation with the production of a single photon is not allowed,
by energy and momentum conservation (Problem 2.4).
The converse processes of pair-production by photons are also possible, and pair-production from a single photon is possible provided
another (charged) particle is present to take up momentum. Quantum
electrodynamics, based on the Dirac and Maxwell equations, describes
all processes involving electrons, positrons and photons to a high degree
of accuracy.
It is a curious fact that nature provides us also with the electrically
charged muon ÿ and tau ÿ and their anti-particles the ‡ and ‡ . Apart
from their greater masses and ®nite lifetimes, muons and taus seem to be
just copies of the electron, and like the electron they are accurately
described by Dirac equations. We shall see that the ÿ can be used as
a probe of nuclear charge density, but otherwise neither the muons nor
the taus play any signi®cant role in nuclear physics.
The remaining leptons are the neutrinos and their corresponding
The experimental evidence (Table 2.1) suganti-neutrinos denoted by .
gests that the mass of a neutrino is certainly very small compared with the
mass of its charged lepton partner. If the mass of a neutrino were zero, it
would, like the photon, travel with the velocity of light.
2.5 The instability of the heavy leptons: muon decay
15
It is exceedingly dif®cult and expensive to carry out experiments with
neutrinos, but there is compelling experimental evidence that the electron,
muon and tau have different neutrinos, e , , associated with them.
2.5
The instability of the heavy leptons: muon decay
The W+ and Wÿ bosons lead to processes called þ-decay, which neither
photons nor Z bosons can induce. In this chapter we illustrate this with
the example of the þ-decay of the muon; in the next chapter we shall
describe þ-decay processes involving hadrons.
The muon decays to a muon neutrino, together with an electron and
an electron anti-neutrino:
ÿ ! ‡ eÿ ‡ e :
The W ®elds play the mediating role in this decay through the two virtual
processes illustrated in Fig. 2.2. Again, in a virtual process actual W
bosons do not appear to an observer.
The W bosons can in principle produce any charged lepton and its
anti-neutrino or an anti-lepton and its neutrino, but energy must be conserved overall. Hence in the case of muon decay the charged lepton must
be an electron. A tau decay can produce a muon or an electron (and
indeed it is suf®ciently massive to decay alternatively to hadrons).
It is of fundamental signi®cance that electric charge is conserved at
every stage of a decay. It is also believed to be true of all interactions that
Fig. 2.2 The decay ÿ ! ‡ eÿ ‡ e . In (a) the muon changes to its neutrino
and a `virtual' Wÿ boson, which then decays to the electron and the electron antineutrino. In (b) a `virtual' W‡ is created from the vacuum with the electron and
the electron anti-neutrino. The W‡ then transforms the muon into a muon neutrino. In these diagrams, the direction of the arrows on the fermion lines follows
the direction of fermion number. (The arrows on anti-particle lines then run
backwards in time.)
16
Leptons and the electromagnetic and weak interactions
to a high degree of approximation a single lepton can only change to
another of the same type, and a lepton and an anti-lepton of the same
type can only be created or destroyed together. There is thus a conservation law, the `conservation of lepton number' (anti-leptons being counted
negatively), for each separate type of lepton. The experimental evidence
for a possible breakdown of this law will be discussed in Chapter 13.
2.6
Parity violation in muon decay
It is observed experimentally that in the decay of the negative muon, the
electron momentum pe is strongly biased to be in the direction opposite to
that of the muon spin s . To explain the implication of this observation
for parity violation, we must ®rst point out that there are two types of
vector.
Under the re¯ection in the origin (Fig. 1.1), the position vector r of a
particle and its momentum p transform:
r ! r0 ˆ ÿr
and
pˆm
dr
ÿdr
! p0 ˆ m
ˆ ÿp;
dt
dt
2:15†
r and p are both true vectors.
The angular momentum L ˆ r p has many of the attributes of a
vector, but under re¯ection
L ! L0 ˆ ÿr† ÿp† ˆ ‡L:
Thus L does not have the re¯ection property (2.14) of the true vectors r
and p. It is called an axial vector or pseudo-vector. The intrinsic angular
momentum s of a particle is likewise an axial vector.
Returning to muon decay, in the re¯ected coordinate system,
pe ! ÿpe , s ! ‡ s , so that the momentum would be said to be biased
in the same direction as the muon spin! It appears that the equations of
the theory are only valid in the original right-handed frame, and would
have to be rewritten to hold in the left-handed re¯ected frame. Thus the
laws are not invariant under re¯ection and hence parity is not conserved
in muon decay. More generally, parity is not conserved in any process
involving the weak interaction ®elds.
The inequivalence of right-handedness and left-handedness is most
extreme in the case of neutrinos. Neutrinos produced in a weak interaction process are always `left-handed', with their spin anti-parallel to their
Problems
17
Fig. 2.3 The relation between momentum p and spin for a neutrino and an
anti-neutrino .
direction of motion, and anti-neutrinos are always `right-handed' (Fig.
2.3). There is no evidence that right-handed neutrinos (or left-handed
anti-neutrinos) exist at all.
The breakdown of parity conservation may be expressed slightly differently. The re¯ection in the origin r ! r0 ˆ ÿr is easily seen to be
equivalent to mirror re¯ection in a plane, followed by a rotation through
about an axis perpendicular to that plane (e.g. the xy-plane and the zaxis, cf. Problem 1.2). There is no evidence that the laws of physics break
down under rotations, so the breakdown is in the mirror re¯ection: the
assumption that the mirror image of a physical process is also a possible
physical process is wrong, in so far as the weak interaction is involved.
Problems
2.1 Plane wave solutions of the relativistic wave-equation for a free particle
of mass m are of the form
ý r; t† ˆ constant†ei krÿ!t†
where
!2 ˆ c2 k2 ‡ m2 c4 = »2 †.
Show that the group velocity of a wave-packet representing a particle of
total energy E ˆ »! is the same as the velocity of a relativistic classical
particle having the same total energy.
2.2 The weak charge density of an electron bound in an atom has a similar
magnitude to the electric charge density and has, similarly, a probability
distribution over the atomic dimensions of the electron's wave-function.
Show that the ratio of the weak interaction energy to the electrostatic
interaction energy between two electrons bound in an atom is of order of
magnitude 4 »= a0 MZ c††2 10ÿ15 , where a0 is the Bohr radius.
(Compare this result with Problem 1.1.)
18
Leptons and the electromagnetic and weak interactions
2.3 An electron-positron pair bound by their Coulomb attraction is called
positronium. Show that when positronium decays from rest to two
photons, the photons have equal energy.
2.4 Use energy and momentum conservation to show that pair annihilation
with the emission of a single photon, e‡ ‡ eÿ ! ÿ, is impossible in free
space.
2.5 Show that a muon in free space with a kinetic energy of 1 MeV will
travel a mean distance of about 90 m before it decays.
2.6 An electron and a ‡ bound by the Coulomb attraction is called
muonium.Which of the following decays can occur?
(a)
‡ eÿ † ! ÿ ‡ ÿ
(b)
‡ eÿ † ! e ‡ (c)
‡ eÿ † ! e‡ ‡ eÿ ‡ e ‡ 2.7 The masses of the electron and neutrinos from a muon decay are negligible compared with the muon mass. Show that if the muon decays
from rest and the kinetic energy released is divided equally between the
®nal leptons then the angle between the paths of any two of them is
approximately 1208.
2.8 Starting from the Coulomb law and the Biot±Savart law, show that the
electric ®eld E is a true vector ®eld, but that the magnetic ®eld B is an
axial vector ®eld.
2.9 Show that a muon is more tightly bound in the lowest state of a 3 H±
muon atom than in the lowest state of a deuteron±muon atom, by about
48 eV. (Note that in the expression m e2 =4"0 †2 =2 »2 for the binding
energy of a hydrogen-like system, m is the reduced mass.) Take the
mass of the deuteron to be 1876 MeV and the mass of the triton 3 H
to be 2809 MeV.
3
Nucleons and the strong interaction
We turn now to the hadrons, bound systems of quarks which interact by
the strong interaction, as well as by the weak and electromagnetic interactions. In particular we shall describe the nucleons, that is to say, the
proton and the neutron, the forces between nucleons, and the effect of the
weak interaction on the stability of nucleons.
3.1
Properties of the proton and the neutron
Nucleons, like leptons, are fermions with spin 12. The mass of the neutron
is 0.14% greater than that of the proton:
mn ˆ 939:566 MeV=c2 ;
mp ˆ 938:272 MeV=c2 :
3:1†
Thus the mass difference mn ÿ mp ˆ 1:29 MeV=c2 ( 2 electron masses).
The neutron has no net electric charge. The proton has the opposite
charge to the electron: protons are responsible for exactly cancelling the
charge of the electrons in electrically neutral atoms.
The electric charge on a proton is not concentrated at a point, but is
symmetrically distributed about the centre of the proton. By the experimental methods to be discussed in Chapter 4, the mean radius Rp of this
charge distribution is found to be Rp 0:8 fm. An extended charge distribution is also found in the neutron, positive charge in the central region
being cancelled by negative charge at greater distances. The matter distribution in nucleons also extends to a distance of about Rp .
19
20
Nucleons and the strong interaction
Both the proton and the neutron have a magnetic dipole moment,
aligned with their spin:
p ˆ 2:792 85 e »=2mp †;
n ˆ ÿ1:913 04 e »=2mp †:
3:2†
Clearly neither magnetic moment is simply related to the value
e »=[2(nucleon mass)] expected from a simple Dirac equation, and this
is a clear indication that the nucleons are not themselves fundamental
particles.
Compelling evidence that the nucleons are the ground states of a
composite system is given by data of which that in Fig. 3.1 is an example.
Fig. 3.1 The total photon cross-section for hadron production on protons
(dashes) and deuterons (crosses). The difference between these cross-sections is
approximately the cross-section on neutrons. (After Armstrong, T. A. et al.
(1972), Phys. Rev. D5, 1640; Nucl. Phys. B41, 445.)
3.2 The quark model of nucleons
21
This shows the cross-section for absorption of photons by protons and by
deuterons (see }3.3), as a function of photon energy up to 1300 MeV. The
cross-sections vary rapidly with energy. A precise de®nition of cross-section is given in Appendix A, but for our immediate purpose it is suf®cient
to remark that the peaks are due to photons being preferentially absorbed
to create an excited state when the photon energy matches the excitation
energy of that state. Perhaps a more familiar example of photons being
absorbed by a composite system is that of atomic absorption. Similar
peaks in atomic absorption cross-sections, but at energies of a few electron volts, correspond to the excitation of the atom to higher energy
states. The nucleon peaks have a similar interpretation, albeit on a very
different energy scale. The ®rst peak in the proton cross-section is at a
photon energy of about 294 MeV, and corresponds to the formation of a
state called the ‡ . The ‡ is a fermion with mass of about
938 ‡ 294† MeV 1232 MeV; its spin has been determined to be 32.
Data for the neutron show that it has a sequence of excited states of
the same spins and almost identical energies as has the proton. The
electrical energies associated with the charge distributions of the proton
and neutron are of order of magnitude e2 = 4"0 Rp † 2 MeV (taking
Rp ˆ 0:8 fm), which is small compared with the nucleon rest mass energies and excitation energies. We shall see that, in all strong interactions,
protons and neutrons behave in the same way to a good approximation.
The near independence of the strong interaction on nucleon type is an
important fact for our understanding of the properties of the nucleus.
3.2
The quark model of nucleons
Any composite system with spin 12 must contain an odd number of fermion constituents. (An even number would give integral spin.) The highly
successful quark model postulates that nucleons contain three fundamental fermions called quarks. We cannot here present the particle physics
which establishes the validity of the quark model, but since particle physics does have implications for the concepts of nuclear physics we give ±
without attempting justi®cation ± some of the most relevant results.
As is the case with the elementary leptons, there are several types of
quark, with a curious and so far unexplained mass hierarchy. For
nucleons and nuclear physics only the two least-massive quarks are
involved, the up quark u and the down quark d. The proton basically
contains two up quarks and a down quark (uud) and the neutron two
downs and one up (ddu). These quarks are bound by the fundamental
22
Nucleons and the strong interaction
strong interaction ®eld, called by particle physicists the gluon ®eld. The
fact that the strong interactions of neutrons are almost the same as those
of protons is explained by the gluon ®eld having the same coupling to all
quarks, independent of their type.
What are the properties of these quarks? They have mass, but the
mass of a particle is generally determined by isolating it and measuring its
acceleration in response to a known force. Because a single quark has
never been isolated, this procedure has not been possible, and our knowledge of the quark masses is indirect. The consensus is that much of the
nucleon mass resides in the gluon force ®elds that bind the quarks, and
only a few MeV/c2 need be assigned to the u and d quark masses. It is well
established that the d quark is heavier than the u quark, since in all cases
where two particles differ only in that a d quark is substituted for a u
quark, the particle with the d quark is heavier. The principal example of
this is the difference in mass between the neutron and proton. The mass,
2 MeV=c2 , associated with the electrical energy of the charged proton is
far greater than that associated with the (overall neutral) charge distribution of the neutron, so that one might expect the proton to be heavier.
However, the extra d quark in the neutron more than compensates for
this, and makes the neutron heavier than the proton.
The electric charges carried by quarks are well veri®ed by measurements of the electromagnetic transitions between the nucleon ground
states and excited states. The u has charge 23e and the d has charge
ÿ 13 e. Thus the proton (uud) has net charge e and the neutron (ddu)
has net charge zero. Again, since a quark has never been isolated, the
evidence for these assignments is all indirect.
The differences between neutrons and protons, other than their electric and weak charges, are due to the u±d mass difference. This has only a
small effect on the basic strong interactions, so that the resulting strong
interaction between nucleons is almost independent of nucleon type. This
independence may be expressed mathematically by introducing the concept of `isotopic spin symmetry', but for our purposes this elaboration is
unnecessary.
3.3
The nucleon±nucleon interaction: the phenomenological
description
We shall see in later chapters that the kinetic energies and potential
energies of nucleons bound together in a nucleus are an order of magnitude smaller than the energies 290 MeV† required to excite the quarks
3.3 The nucleon±nucleon interaction: the phenomenological description
23
in an individual nucleon. It is, therefore, reasonable to regard a nucleus as
an assembly of nucleons interacting with each other, but basically remaining in their ground states. To understand the physics of nuclei it is therefore important to be able to describe the interactions between nucleons.
Since nucleons are composite particles, we can anticipate that their interactions with each other will not be simple. In fact they are rather complicated. Nevertheless, after 70 years of experimental and theoretical
effort a great deal is known empirically about the forces between two
nucleons, especially at the low energies relevant to nuclear physics.
The empirical approach is to construct a possible potential which
incorporates our limited theoretical knowledge (which we shall discuss
in }3.4) and has adjustable features, mainly to do with the short-range
part of the interaction. The SchroÈdinger equation for two nucleons interacting through this potential is then solved numerically and the adjustable
features are varied to ®t the experimental facts, namely the properties of
the deuteron and the low-energy scattering data.
The deuteron is a neutron±proton bound state with:
binding energy = 2.2245 MeV;
angular momentum j ˆ 1;
magnetic moment = 0.8574 e »=2mp †;
3:3†
electric quadrupole moment = 0.286 fm2 :
Neither proton±proton nor neutron±neutron bound states exist.
The scattering data provide much more information. Nucleons have
spin 12, which may be `¯ipped' in the scattering. It can be shown that there
are ®ve independent differential cross-sections for spin-polarised proton±
proton and a further ®ve for proton±neutron scattering which can, in
principle, be measured. Neutron±neutron cross-sections have never
been measured directly because there are no targets of free neutrons.
As has been explained, the strong neutron±neutron interaction should
be almost the same as the strong proton±proton interaction, and both
these should be almost the same as the proton±neutron interaction for the
same states of relative motion. However, we must remember here the
Pauli exclusion principle: the neutron and proton can exist together in
states which are not allowed for two protons or two neutrons. This is why
the neutron and proton can have a bound state, whereas two protons or
two neutrons do not bind, without any contradiction of the principle that
the strong interaction is almost independent of nucleon type.
24
Nucleons and the strong interaction
A large amount of careful and accurate data has been accumulated,
and the most sophisticated and accurate empirical potential has been
constructed by a group of scientists working in Paris. Two expressions
are needed: one for the (anti-symmetric) states allowed for two protons or
two neutrons, as well as a proton and a neutron, and one for symmetric
states accessible only to the neutron±proton system. For both cases, when
the spins of the two nucleons are coupled to give a total spin S ˆ 0 (see
Appendix C) the nucleons only experience a central potential.
In Fig. 3.2, the central potential for the anti-symmetric states with
S ˆ 0 is denoted by VC0 . The central potential for symmetric states differs from this, and is not shown.
When the spins couple to S ˆ 1 there are four contributions to these
potentials, which are then each of the form
V r† ˆ VC1 r† ‡ VT r†ÿT ‡ VSO r†ÿSO ‡ VSO2 r†ÿSO2 ;
where
r1 r† r2 r†
ÿ r1 r2
r2
ˆ r1 ‡ r2 † L
ÿT ˆ 3
»ÿSO
3:4†
»2 ÿSO2 ˆ r1 L† r2 L† ‡ r2 L† r1 L†:
In these expressions r »=2† is the nucleon spin operator, L is the orbital
angular momentum operator of the nucleon pair, and the subscripts 1 and
2 refer to the two nucleons present.
Again, the radial factors differ in the two cases of symmetric and antisymmetric states. In Fig. 3.2, VC1 , VSO and VT correspond to symmetric
states.
VC1 is essentially an ordinary central potential. VT ÿT is called the
tensor potential. It has the same angular structure as the potential
between two magnetic dipoles and it is also interesting because it is the
only part of the potential which does not commute with L2 , so that l is
not a good quantum number. VSO ÿSO and VSO2 ÿSO2 give rise to different
terms for the different couplings of spin and orbital angular momenta.
Spin±orbit coupling is well known in atomic physics, where it is due to
magnetic interactions. However, these terms in the nuclear potential,
which are of major importance, arise out of the strong interaction.
3.3 The nucleon±nucleon interaction: the phenomenological description
25
Fig. 3.2 The most important components of the `Paris potential'. (After
Lacombe, M. et al. (1980), Phys. Rev. C21, 861.)
In Fig. 3.2 we show the four potentials that are most important at low
energies of interaction (< 100 MeV) and in particular are important for
nucleons in nuclei.
The potential VCO r† is appropriate for low-energy proton±proton
and neutron±neutron interactions. The attractive tail is not, however,
suf®ciently deep to bind two nucleons. The potentials VC1 r†, VSO r†
and VT r† are responsible for binding the deuteron: note the deeply
attractive part of VT r†, which is associated also with the large electric
quadrupole moment of the deuteron.
The tensor potential is particularly important for binding the deuteron, but since it is zero on averaging over all directions it becomes less
important in heavier nuclei. This last remark presupposes that the poten-
26
Nucleons and the strong interaction
tial established for the interaction of two nucleons in isolation is relevant
when many nucleons are interacting in an atomic nucleus. We shall discuss this assumption further in Chapter 4.
For the moment, we simply note the similarity between the central
potentials and the well-known Lennard-Jones pair potential between neutral atoms, which also has a repulsive core and attractive tail, and (albeit
on a different scale) binds the atoms together in condensed matter.
3.4
Mesons and the nucleon±nucleon interaction
Like all fermions, quarks have corresponding anti-particles. Anti-protons
and d d u†;
the
and anti-neutrons can exist, made up of anti-quarks, u u d†
excited states of nucleons have images of identical mass but opposite
charge in anti-quark matter. In fact the electromagnetic and strong interactions of anti-matter seem to be identical to those of matter. It is possible
to contemplate the existence of stable anti-atoms, and macroscopic
bodies, made up of anti-matter, but as electrons annihilate with positrons,
so do nucleons annihilate with anti-nucleons; matter and anti-matter,
though stable in isolation, cannot coexist. To study anti-particles we
must create them in laboratories.
As well as binding three quarks or three anti-quarks together to make
nucleons and anti-nucleons, the strong gluon ®eld can bind a quark and
an anti-quark together to form a short-lived particle called a meson. Like
nucleons, such bound pairs have a sequence of excited states.
Of most importance for nuclear physics are the mesons. The elec and du†
trically charged ‡ and ÿ are made up of ud†
p pairs respec0
tively, and the neutral is a superposition uu ÿ dd†= p2 of quark anti
quark pairs. (The orthogonal combination uu ‡ dd†=
2 belongs to a
meson called the .)
The masses of the mesons are:
mass of ‡ ˆ mass of ÿ ˆ 139:57 MeV=c2 ;
mass of 0 ˆ 134:9 MeV=c2 :
3:5†
(The has mass 547 MeV=c2 .)
The quark±anti-quark pairs in these mesons have orbital angular
momentum zero and intrinsic spins coupled to give total angular momentum zero. The ®rst excited states also have orbital angular momentum
zero, but the intrinsic spins are coupled to give a total spin with quantum
3.4 Mesons and the nucleon±nucleon interaction
27
number S ˆ 1. These states are called the ‡ , ÿ and 0 mesons; they
have masses 770 MeV=c2 .
Quarks are sources of the gluon ®eld, and in a nucleon they are
con®ned by this ®eld to lie within the nucleon. At distances 1 fm the
force between nucleons is not mediated by the basic gluon ®eld, but rather
by the exchange of mesons. Mesons have integral spin and are bosons, as
are the photons which mediate the electromagnetic interaction and the
W and Z particles which mediate the weak interaction.
Although mesons are composite particles, their motion as a whole is
still described by a wave-function r; t†, obeying in free space the waveequation for massive particles:
"
#
1 @2 mc2
r ÿ 2 2ÿ
r; t† ˆ 0;
»
c @t
2
3:6†
where m is the mass of the particle (cf. equations (2.10)±(2.12)).
One solution of this equation describes the meson ®eld associated
with a nucleon having operator spin r1 »=2† at r1 :
r; t† ˆ g r1 ;1 †
eÿmcjrÿr1 j= »
;
jr ÿ r1 j
3:7†
where g is a measure of the meson source strength of the nucleon. The
gradient operator ;1 acts only on r1 , so that (3.7) is evidently a solution of
(3.6) (cf. (2.11)).
The ®eld (3.7) changes sign under re¯ection in the origin (see }2.6)
and is said to be a pseudo-scalar ®eld. It is the simplest such ®eld we can
construct which satis®es the wave-equation. The `dipole-like' nature of
the ®eld is well understood by particle physicists, and the interaction
energy between two nucleons associated with it is of `dipole±dipole' form:
U12 / g2 r2 ;2 † r1 ;1 †
eÿmcjr2 ÿr1 j= »
:
jr2 ÿ r1 j
3:8†
The mesons are the mesons of smallest mass and hence give the
largest contribution to the interaction at large distances. The appropriate
length scale, from the exponential in (3.7), is
»=mc 1:4 fm:
28
Nucleons and the strong interaction
Explicit differentiation shows that (3.8) includes a potential of the
tensor form VT r†ÿT . It is well established that meson exchange is
responsible for most of the tensor potential (3.4), and is the dominant
contribution to the whole potential at distances jr2 ÿ r1 j > 1:4 fm. At
smaller distances other meson exchange processes become important,
including the exchange of mesons. However, the potentials at distances
< 0:8 fm and, in particular, the short-range repulsion, are empirical and
so far have no established explanation.
3.5
The weak interaction: þ-decay
Hadrons are subject to the weak interaction as well as to the electromagnetic and strong interactions, and it is through the weak interaction that
quarks, like leptons, are coupled to the W and Z bosons. For example,
one quark can change to another by emitting or absorbing a virtual W
boson. The phenomena of þ-decay, in which a neutron becomes a proton
or a proton becomes a neutron, proceed in this way.
In free space, the proton is the only stable three-quark system.The
neutron in free space has enough excess mass over the proton to decay to
it by the process shown in Fig. 3.3.
The mean life of the neutron in free space is 886:7 s 15 minutes.
However, a neutron bound in a nucleus will be stable if the nuclear
binding energies make decay energetically forbidden. Conversely, a proton bound in a nucleus may change into a neutron
p ! n ‡ e ‡ ‡ e ;
if the nuclear binding energies involved allow the process to occur. The
energetics of þ-decay will be dealt with in detail in Chapter 4, and a more
quantitative theory of þ-decay will be given in Chapter 12.
Fig. 3.3 The decay n ! p ‡ eÿ ‡ e . As with muon decay, parity is not conserved in this weak interaction.
3.6 More quarks
29
Table 3.1. Properties of quarks
Quark
Approximate mass
Electric charge (e)
Down d
Up u
Strange s
Charm c
Bottom b
Top t
3±9 MeV/c2
1.5±5 MeV/c2
60±170 MeV/c2
1.1±1.4 GeV/c2
4.1±4.4 GeV/c2
174 GeV/c2
ÿ 13
3.6
2
3
1
ÿ3
2
3
1
ÿ3
2
3
More quarks
The u and d quarks are merely the two least massive of a sequence of
types, or `¯avours' of quark, and to set the discussion of þ-decay above
into this wider context we list in Table 3.1 all the presently known ¯avours.
The existence of the more massive quarks in this table is revealed by
the observation of states similar to the nucleon states and meson states we
have already discussed, but which are apparently formed by substituting
any of the `new' quarks for the u or d quarks. Thus, for example, substituting an s quark for a d quark, there exists a K+ meson us† (mass
but heavier, and a 0 baryon
493.68 MeV/c2 ) like the ‡ meson ud†
2
(uds) (mass 1193 MeV/c ) like the neutron (udd) but heavier. Baryon
and anti-baryon are the generic names for particles essentially made up
of three quarks or three anti-quarks. Again, since no quark has ever been
isolated, the masses given in Table 3.1 are effective masses and have no
precise signi®cance.
Were it not for the weak interaction a heavy quark would be stable
and there would be more absolute conservation laws, for example, the
conservation of strangeness and the conservation of charm. Such laws
hold for processes involving only the electromagnetic and strong interactions, but are not absolute since all quarks couple to the W and Z weak
interaction ®elds, and a quark changes its ¯avour (but remains a quark!)
when it emits or absorbs a virtual W boson. Thus, for example, the s
quark in the ÿ baryon can decay through processes like those shown in
Fig. 3.4. We shall see that nuclear binding energies are not suf®ciently
large to make a baryon containing a heavy quark stable even in a nucleus.
The weak interaction makes all mesons unstable. Mesons containing
a heavy quark can decay by the heavy quark changing into a lighter
30
Nucleons and the strong interaction
Fig. 3.4 The decays ÿ ! n ‡ ÿ , ÿ ! n ‡ ÿ ‡ .
quark. Another possible process is illustrated in Fig. 3.5, in which a quark
and an anti-quark annihilate through the weak interaction into an antimuon and a muon neutrino. This latter process is the predominant type of
decay of the charged pions. The mean life of charged pions is
2:60 10ÿ8 s.
The 0 usually decays into two photons by the direct annihilation of
the quarks with their own anti-quarks, in a way rather similar to the
decay of positronium (an electron±positron pair e‡ eÿ in a bound state).
Such a decay (Fig. 3.6) takes place through the electromagnetic interaction, and is therefore much quicker: the mean life of the 0 is
0:84 10ÿ16 s.
All the available experimental evidence is consistent with there being
a law of `conservation of baryon number': the total number of baryons
(anti-baryons being counted negatively) is conserved in all interactions, so
that a baryon and an anti-baryon are always created or destroyed
Fig. 3.5 The decay ‡ ! ‡ ‡ . The charged pion was discovered by Powell
and co-workers in Bristol in 1947 by the observation of this decay.
Fig. 3.6 The electromagnetic decay 0 ! ÿ ‡ ÿ.
3.7 The Standard Model of particle physics
31
together. Indeed, it has been established that a lower limit to the mean life
of an isolated proton exceeds 1:6 1025 years.
3.7
The Standard Model of particle physics
The electromagnetic, weak, and strong interactions of leptons and quarks
are combined in the theoretical edi®ce known as the Standard Model of
particle physics. This model has been remarkably successful in the interpretation of the data of particle physics. It is generally believed that the
properties of nuclei, and the phenomena of nuclear physics in general, are
a consequence of the established laws of particle physics. Our presentation of nuclear physics has been guided by the Standard Model, but a
detailed understanding of, even, the proton within the Standard Model
remains an experimental and theoretical challenge.
The concepts which are useful at the low energies we consider, were
developed long before the Standard Model was established. We shall see
in the following chapters that quite simple theoretical models are highly
successful in elucidating the properties of nuclei.
Problems
3.1 The spins of the neutron and the proton in the deuteron are aligned.
Show that the magnetic moment of the deuteron is within 3% of the
sum of the neutron and proton moments. What might be the origin of
the discrepancy?
3.2(a) Show that the magnetic interaction energy between two magnetic dipoles
r1 and r2 is of the form VT r†ÿT with VT r† ˆ ÿ 0 =4†2 =r3 . (0 is
the permeability of the vacuum.)
(b) Verify that equation (3.8) includes terms in the nucleon±nucleon potential of tensor form.
3.3 The Coulomb self-energy of a hadron with charge ‡e or ÿe is about
1 MeV. The quark content and rest energies (in MeV) of some hadrons
are:
n(udd) 940, p(uud) 938
ÿ (dds) 1197, 0 (uds) 1192, ‡ (uus) 1189
K0 ds† 498, K‡ us† 494.
The u and d quarks make different contributions to the rest energy.
Estimate this difference.
3.4 Which of the following processes are allowed by the conservation laws?
(a) n ! p ‡ ÿ,
32
Nucleons and the strong interaction
(b) p ! e‡ ‡ ÿ,
(c) p ! ‡ ‡ ÿ,
(d) p ‡ n ! ÿ ‡ 0 .
3.5 The decay of the ÿ initiates the sequence of decays shown below:
ÿ ! Kÿ ‡ 0
ÿ!0 ‡ ÿ
ÿ!p ‡ eÿ ‡ e
ÿ
ÿ! ‡ 0
ÿ!ÿ ‡ ÿ
ÿ!ÿ ‡ ÿ!eÿ ‡ e ‡ The quark content of the hadrons involved is:
ÿ (ssd), 0 (sud), 0 (sud), p(uud),
ÿ du†,
0 uu ÿ dd†.
Kÿ s; u†,
Classify the decays as strong, electromagnetic, or weak.
4
Nuclear sizes and nuclear masses
We now begin our study of the nucleus. A nucleus is a bound assembly of
neutrons and protons. A
Z X denotes a nucleus of an atom of the chemical
element X containing A nucleons, of which Z are protons and
N ˆ A ÿ Z† are neutrons. For example, 35
17 Cl denotes a chlorine nucleus
Cl
a
chlorine
nucleus
with 20 neutrons. Since the
with 18 neutrons and 37
17
chemical symbol determines the atomic number Z, 35 Cl or 37 Cl is identi®cation enough, but the addition of the Z label is often useful.
A ˆ N ‡ Z† is called the mass number of the nucleus. Nuclei which differ
only in the number of neutrons they contain are called isotopes. Nuclei of
the same A but different Z are called isobars.
4.1
Electron scattering by the nuclear charge distribution
Rutherford's famous analysis in 1911 of the scattering of -particles by
matter established that the size of the nucleus of an atom is small compared with the size of the atom. Whereas the electronic distribution
extends to a distance of the order of aÊngstroÈms 1 A ˆ 10ÿ10 m† from
the nucleus, these and later experiments showed that the distribution of
nucleons is con®ned to a few fermis 1 fm ˆ 10ÿ15 m†. Early theories of decay and nuclear binding energies gave estimated values for nuclear radii
of a similar magnitude.
Precise information came in the 1950s, with experiments using the
elastic scattering of high-energy electrons to probe the nuclear charge
distribution. There is an obvious advantage in using charged leptons
(electrons or muons) to probe nuclear matter, since leptons interact
with nucleons primarily through electromagnetic forces: the complica33
34
Nuclear sizes and nuclear masses
tions of the strong nuclear interaction are not present, and the weak
interaction is negligible for the scattering process. The most signi®cant
interaction between a charged lepton, which can be regarded as a structureless point object, and the nuclear charge is the Coulomb force, and
this is well understood. If the nucleus has a magnetic moment, the magnetic contribution to the scattering becomes important at large scattering
angles, but this also is well understood.
If scattering experiments are to give detailed information on the
nuclear charge distribution, it is clear that the de Broglie wavelength of the incident particle must be less than, or at least comparable with, the
distances over which the nuclear charge density changes. An electron with
=2† 1 fm has momentum p ˆ 2»= and hence energy
1
E ˆ p2 c2 ‡ m2 c4 †2 200 MeV. At these energies, the electrons are
described by the Dirac relativistic wave-equation, rather than by the
SchroÈdinger equation. The experiments yield a differential cross-section
d E; †=dÿ (Appendix A) for elastic scattering from the nucleus through
an angle , which depends on the energy E of the incident electrons.
Typical experimental data are shown in Fig. 4.1.
The incident electrons are, of course, also scattered by the atomic
electrons in the target. However, this scattering is easily distinguished
from the nuclear scattering by the lower energy of the scattered electrons.
Whereas the recoil energy taken up by the heavy nucleus is very small, the
recoil energy taken up by the atomic electrons is appreciable, except for
scattering in the forward direction. (See Problem 4.1.)
The nuclear charge density will be described by some density function
ech r†. (The proton charge e is put in as a factor for convenience.) This
function is not necessarily spherically symmetric ± we shall mention this
later ± but for nuclei which are spherically symmetric, or nearly so, we can
assume the charge density depends only on the distance r from the centre
of the nucleus. Then, using the Dirac wave-equation for the electron,
d=dÿ is in principle completely determined by ch r†, though the calculations are not trivial. The inverse problem, that of ®nding ch r† from a
knowledge of d=dÿ, is even more dif®cult (see Problem 4.2). The
restricted amount of experimental information available means that, at
best, only a partial resolution of the problem can be made. Some idea of
the results of a direct inversion of scattering data is given by Fig. 4.2.
It has been more usual to assume a plausible shape for ch r†, describe
this by a simple mathematical expression involving a few parameters, and
then determine the parameters by ®tting to the scattering data. A form
which has been widely adopted is
4.1 Electron scattering by the nuclear charge distribution
35
Fig. 4.1 Experimental elastic electron-scattering differential cross-section from
gold 197
79 Au at energies of 126 MeV and 183 MeV. The ®tted curves are calculated
with an assumed charge distribution of the form given by equation (4.1), with
R ˆ 6:63 fm, a ˆ 0:45 fm. The cross-section to be expected, at 126 MeV, if the
gold nucleus had a point charge is shown for comparison. (Data and theoretical
curves taken from Hofstadter, R. (1963), Electron Scattering and Nuclear and
Nucleon Structure, New York: Benjamin.)
ch r† ˆ
0ch
;
1 ‡ e rÿR†=a
4:1†
where the parameters to be determined are R and a, and 0ch is a normalisation constant chosen so that
Z
ch r†d3 r ˆ 4
Z
0
1
ch r†r2 dr ˆ Z:
It should be stressed that the choice of this expression has no fundamental
signi®cance, it just conveniently describes a charge distribution which
extends almost uniformly from the centre of the nucleus to a distance
R, and falls to zero over a well-de®ned surface region of thickness a.
This picture is consistent with the results of direct inversion.
36
Nuclear sizes and nuclear masses
Fig. 4.2 The electric charge density of 208
82 Pb from a model-independent analysis
of electron scattering data. The bars indicate the uncertainty. (Friar, J. L. &
Negele, J. W. (1973), Nucl. Phys. A212, 93.)
In Fig. 4.3 we show nuclear charge distributions for a light 168 O†, a
208
medium 109
47 Ag† and a heavy 82 Pb† nucleus obtained from experimental
scattering data, using this parametrisation of the charge density. The
corresponding values of R and a are given in Table 4.1.
As the examples in the table indicate, it appears that there is a wellde®ned `surface region' which has much the same width for all nuclei,
even light ones.
4.2
Muon interactions
The negative muon is another leptonic probe of nuclear charge. Its properties, other than its mass of m 207 me and its mean life of
Table 4.1. Nuclear radii (R) and nuclear surface widths (a)
1
Nucleus
R
(fm)
a
(fm)
R=A3
(fm)
16
8O
109
47 Ag
208
82 Pb
2.61
5.33
6.65
0.513
0.523
0.526
1.04
1.12
1.12
4.3 The distribution of nuclear matter in nuclei
37
Fig. 4.3 The electric charge density of three nuclei as ®tted by
ch r† ˆ 0ch =‰1 ‡ exp r ÿ R†=a†Š. The parameters are taken from the compilation in Barrett, R. C. & Jackson, D. F. (1977), Nuclear Sizes and Structure,
Oxford: Clarendon Press.
2:2 10ÿ6 s, are similar to those of the electron. However, the radius of
its lowest Bohr orbit in an atom of charge Z is 4"0 † »2 =m Ze2 , and this
is smaller than the corresponding electron orbit by a factor me =m †. For
Z ˆ 50 the radius is only 5 fm. Hence the wave-functions of the lowest
muonic states will lie to a considerable extent within the distribution of
nuclear charge, particularly in heavy nuclei, and the energies of these
states will therefore depend on the details of the nuclear charge distribution.
Experimentally, negative muons are produced in the target material
by the decay of a beam of negative pions, and are eventually captured in
outer atomic orbitals. Before they decay, many muons fall into lower
orbits, emitting X-rays in the transitions. The measured energies of
these X-rays may be compared with those calculated with various choices
of parameters for ch r†. Values of R and a, found in this way, agree well
with results from electron scattering.
4.3
The distribution of nuclear matter in nuclei
From the distribution of charge in a nucleus, which as we have seen can
be determined by experiment, we can form some idea of the distribution
38
Nuclear sizes and nuclear masses
of nuclear matter. If the proton were a point object, we could identify the
proton number density p r† with ch r†. Since the strong nuclear forces
which bind nucleons together are charge independent and of short range,
we can assume that to a good approximation the ratio of neutron density
n to proton density p is the same at all points in a nucleus, i.e.
n r†=p r† ˆ N=Z. Then the total density of nucleons ˆ n ‡ p can
be expressed as ˆ A=Z†ch , where A ˆ N ‡ Z. The resulting nuclear
matter densities for the same nuclei we took in Fig. 4.3 are plotted in Fig.
4.4. These densities are only approximate, since we have neglected the
®nite size of both proton and neutron and the effect of Coulomb forces,
but they indicate that at the centre of a nucleus the nuclear matter density
is roughly the same for all nuclei. It increases with A, but appears to
tend to a limiting value 0 of about 0.17 nucleons fmÿ3 for large A. The
existence of this limiting value 0 , known as the `density of nuclear matter', is an important result. Consistently with this, we ®nd (Table 4.1),
1
that the `radius' R of a nucleus is very closely proportional to A3 , and,
3
approximately, 4=3†R 0 ˆ A. We shall take
0 ˆ 0:17 nucleons fmÿ3
4:2†
Fig. 4.4 The nucleon density of the nuclei of Fig. 4.3, with r† ˆ A=Z†ch r†.
4.4 The masses and binding energies of nuclei in their ground states
39
which implies
1
R ˆ 1:12 A3 fm:
4.4
The masses and binding energies of nuclei in their ground
states
It thus appears that a nucleus is rather like a spherical drop of liquid, of
nearly uniform density. How are we to understand its properties? A
nucleus is a quantum-mechanical system. We shall see later that its
excited states are generally separated by energies 1 keV or more from
its ground state, so that to all intents and purposes nuclei in matter at
temperatures that are accessible on Earth are in their ground states. Like
any other ®nite system, a nucleus in its ground state has a well-de®ned
energy and a well-de®ned angular momentum. In this chapter we shall be
concerned with the ground-state energy. Other ground-state properties of
a nucleus will be discussed in the next chapter.
Since a nucleus is a bound system, an energy B Z; N† is needed to pull
it completely apart into its Z protons and N neutrons. From the Einstein
relation between mass and energy, the binding energy B Z; N† is related to
the mass mnuc Z; N† of the nucleus by
mnuc Z; N† ˆ Zmp ‡ Nmn ÿ B Z; N†=c2 ;
4:3†
and B Z; N† must be positive for the nucleus to be formed. We shall see
that nuclear binding energies are of the order of 1% of the rest-mass
energy mnuc c2 .
Experimentally, the masses of atomic ions, rather than the masses of
bare nuclei, are the quantities usually measured directly. If ma Z; N† is the
mass of the neutral atom,
ma Z; N† ˆ Z mp ‡ me † ‡ Nmn ÿ B Z; N†=c2 ÿ belectronic =c2 ;
4:4†
where belectronic is the binding energy of the atomic electrons. These electronic contributions are, for many purposes, negligible. (The simple
Thomas±Fermi statistical model of a neutral atom gives the total electro7
nic binding energy 20:8Z 3 eV.)
Atomic masses are known very accurately, and published tables give
atomic masses rather than nuclear masses. Measurements in `mass spec-
40
Nuclear sizes and nuclear masses
trometers' depend on the de¯ection of charged ions in electric and magnetic ®elds. Instruments of great ingenuity have been developed, giving
relative masses accurate to about one part in 107 . The unit employed is
1
the atomic mass unit, which is de®ned to be 12
of the mass of the neutral
12
C atom:
1 amu ˆ 931:494 32 0:000 28 MeV=c2 :
Differences between the masses of stable atoms and unstable, radioactive, atoms (for which mass spectrometers may be inappropriate) can
be determined by measuring the energy release in the unstable atom
decay, again using the Einstein mass±energy relation.
Table 4.2 shows the experimental binding energies for some of the
lighter nuclei, those formed by successively adding a proton followed by a
neutron to an original neutron. Note that all the binding energies are
positive: this re¯ects the basic long-range attraction of the nucleon±
nucleon interaction.
Also given in the table is the average binding energy per nucleon,
B Z; N†=A. For the heavier nuclei in the table, the average binding energy
appears to be gradually increasing to around 8 MeV, but the numbers
¯uctuate somewhat from nucleus to nucleus. The ¯uctuation is more
dramatically exhibited in the binding energy difference between a nucleus
and the one preceding it, also shown in the table. This energy can be
interpreted as the binding energy of the last nucleon added to the nucleus
in the given sequence. It is particularly large for the `even±even' nuclei
4
8
12
16
2 He, 4 Be, 6 C and 8 O, and particularly small for the nuclei immediately
following, growing steadily as the next three nucleons are added to form
the next even±even nucleus. Clearly we see here some extra binding
energy associated with neutron±neutron and proton±proton pairing.
The effect stems from the attractive character of the nucleon±nucleon
interaction, and is associated with the pairing of angular momenta
which will be discussed in Chapter 5. Table 4.2 also gives the spins and
parities of the nuclei for later reference; it will be seen that the even±even
nuclei have spin zero.
As we shall see in Chapter 6, because of its low mass, low electric
charge, and relatively large binding energy, the ®rst even±even nucleus
4
2 He is particularly important in the nuclear physics of heavy nuclei.
Indeed, 42 He played an important role in the early history of nuclear
physics and before it was properly identi®ed it was given a special
name, the -particle, a name still in use today.
4.5 The semi-empirical mass formula
41
Table 4.2. Energies of some light nuclei
Nucleus
Binding
energy
(MeV)
Binding
energy of
last nucleon
(MeV)
Binding
energy per
nucleon
(MeV)
2
1H
3
2H
4
2 He
5
2 He
6
3 Li
7
3 Li
8
4 Be
9
4 Be
10
5B
11
5B
12
6C
13
6C
14
7N
15
7N
16
8O
17
8O
2.22
8.48
28.30
27.34
31.99
39.25
56.50
58.16
64.75
76.21
92.16
97.11
104.66
115.49
127.62
131.76
2.2
6.3
19.8
ÿ1:0
4.7
7.3
17.3
1.7
6.6
11.5
16.0
5.0
7.6
10.8
12.1
4.1
1.1
2.8
7.1
5.5
5.3
5.6
7.1
6.5
6.5
6.9
7.7
7.5
7.5
7.7
8.0
7.8
Spin and
parity
1‡
1‡
2
‡
0
3ÿ
2
‡
1
3ÿ
2
‡
0
3ÿ
2
‡
3
3ÿ
2
‡
0
1ÿ
2
‡
1
1ÿ
2
‡
0
5‡
2
Some of the large binding energies of the nuclei 42 He, 126 C and 168 O can
be associated with their `shell structure', which will be discussed in
Chapter 5. As for 84 Be, its binding energy is less than that of two particles by 0.1 MeV, and so the nucleus 84 Be is unstable. It does have a
transient existence for a long time compared with the `nuclear time-scale'
(}5.2), but if it is formed it will eventually fall apart into two -particles.
Another interesting special case in Table 4.2 is that of 52 He. The
binding energy of the last nucleon is here negative; if 52 He is formed it,
too, has only a transient existence before falling apart into a neutron and
an -particle. The other nuclei in Table 4.2 are all stable.
4.5
The semi-empirical mass formula
The features of `pairing energies' and shell-structure effects, superposed
on a slowly varying binding energy per nucleon, can be discerned
throughout the range of nuclei for which data are available. We saw in
42
Nuclear sizes and nuclear masses
}4.3 that the density of nuclear matter is approximately constant, and also
that nuclei have a well-de®ned surface region. It appears as if a nucleus
behaves in some ways rather like a drop of liquid. This analogy is made
more precise in the `semi-empirical mass formula', a remarkable formula
which, with just a few parameters, ®ts the binding energies of all but the
lightest nuclei to a high degree of accuracy. There are several versions of
the mass formula. The one which is suf®ciently accurate for the purposes
of this book gives for the total binding energy of a nucleus of A nucleons,
made up of Z protons and N neutrons,
2
B N; Z† ˆ aA ÿ bA3 ÿ s
N ÿ Z†2 dZ 2
ÿ 1 ÿ 1:
A
A2
A3
4:5†
The parameters a, b, s, d and are found by ®tting the formula to
measured binding energies. Wapstra (Handbuch der Physik, XXXVIII/
1) gives
a = 15.835 MeV
b = 18.33 MeV
s = 23.20 MeV
d = 0.714 MeV
and
(
=
+11.2 MeV for odd±odd nuclei (i.e., odd N, odd Z)
0 for even±odd nuclei (even N odd Z, or even Z, odd N)
ÿ11.2 MeV for even±even nuclei (even N, even Z).
It is the ®rst two terms in this formula which have an analogue in the
theory of liquids. The term aA† represents a constant bulk-binding
energy per nucleon, like the cohesive energy of a simple liquid. The second term represents a surface energy, in particular the surface energy of a
sphere. The surface area of a sphere is proportional to the two-thirds
2
power of its volume and hence, at constant density of nucleons, to A3 .
As in a liquid, this term subtracts from the bulk binding since the particles
in the surface are not in the completely enclosed environment of those in
the bulk. In liquids this term is identi®ed with the energy of surface
tension, and is responsible for drops of liquid being approximately spherical when gravitational effects are small. In nuclei, gravitational effects
are always small, and indeed nuclei do tend to be spherical.
1
The term ÿdZ2 =A3 , called the Coulomb term, also has a simple
explanation; it is the electrostatic energy of the nuclear charge distribu-
4.5 The semi-empirical mass formula
43
1
tion. If the nucleus were a uniformly charged sphere of radius R0 A3
(equation (4.2)) and total charge Ze, it would have energy
Ec ˆ
3
Ze†2
_
5 4"0 †R0 A13
4:6†
With R0 ˆ 1:12 fm this gives an estimate of d, d ˆ 0:78 MeV, close to the
value found empirically.
The term ÿs N ÿ Z†2 =A is the simplest expression which, by itself,
would give the maximum binding energy, for ®xed A, when N ˆ Z (A
even) or N ˆ Z 1 (A odd). It is called the symmetry energy, since it
tends to make nuclei symmetric in the number of neutrons and protons. As was exempli®ed in the case of the deuteron discussed in
Chapter 3, the average neutron±proton attraction in a nucleus is
greater than the average neutron±neutron or proton±proton attraction,
essentially as a consequence of the Pauli exclusion principle. Thus for a
given A it is energetically advantageous to maximise the number of
neutron±proton pairs which can interact: this is achieved by making Z
and N as near equal as possible. Since the forces are short range, the
term must correspond to a `bulk' effect, like the cohesive energy.
Hence there must be a factor A in the denominator, so that overall
the term is proportional to A for a ®xed ratio of neutrons to protons.
One can also argue (see Problem 5.2) that the kinetic energy contribution to the energy results in a similar term, which is absorbed in the
coef®cient s.
The ®nal term in the semi-empirical mass formula is the pairing
1
energy =A2 , manifest in the light nuclei included in Table 4.2. It is purely
1
phenomenological in form and the Aÿ2 dependence is empirical. For the
larger nuclei the pairing energy is small but, as we shall see, it does give
rise to important physical effects.
More sophisticated versions of the formula include also `shell structure' effects (Chapter 5), but for nuclei heavier than neon A ˆ 20† for
which our formula is appropriate these extra terms are of less signi®cance
than the ®ve terms of equation (4.5).
We have in the semi-empirical mass formula a description and an
understanding of the binding energies of the nuclei. We shall see that it
gives a simple but profound explanation of the masses of the chemical
elements and of why there is only a ®nite number of stable atoms in
chemistry.
44
Nuclear sizes and nuclear masses
4.6
The ÿ-stability valley
Using equations (4.3) and (4.5), the mass of the neutral atom with its
nucleus having Z protons and N neutrons is given by
2
ma N; Z†c2 ˆ Nmn ‡ Z mp ‡ me ††c2 ÿ aA ‡ bA3 ‡
s N ÿ Z†2
‡ 1;
‡
A
A2
dZ 2
1
A3
4:7†
(neglecting the electron binding energies).
For a ®xed number of nucleons A, we can write this as a function of
Z, replacing N by A ÿ Z:
2
1
ma A; Z†c2 ˆ Amn c2 ÿ aA ‡ bA3 ‡ sA ‡ Aÿ2 †
1
ÿ 4s ‡ mn ÿ mp ÿ me †c2 †Z ‡ 4sAÿ1 ‡ dAÿ3 †Z 2
ˆ
ÿ ÿZ ‡ þZ 2 ;
say:
4:8†
Consider ®rst the case A odd, so that ˆ 0. The plot of ma A; Z† against
Z is a parabola, with a minimum at
Z ˆ ÿ=2þ ˆ
4s ‡ mn ÿ mp ÿ me †c2 †A
2
2 4s ‡ dA3 †
:
4:9†
Thus the atom with the lowest rest-mass energy for given A has Z equal
to the integer Zmin closest to ÿ=2þ. From the form of the expression (4.9)
and the values of the parameters, it is evident that Zmin 4 A=2, so that
N 5 Z for this nucleus.
Now ÿ-decay, described in }3.5, is a process whereby the Z of a
nucleus changes while A remains ®xed, if the process is energetically
allowed. Thus if a nucleus has Z < Zmin the process
A; Z† ! A; Z ‡ 1† ‡ eÿ ‡ e
is possible if
mnuc A; Z† > mnuc A; Z ‡ 1† ‡ me ;
4:10†
4.6 The ÿ-stability valley
45
since the mass of the anti-neutrino (if indeed it has mass) is exceedingly
small. Adding Zme to each side of this inequality, the condition may be
written in terms of atomic masses:
ma A; Z† > ma A; Z ‡ 1†:
4:11†
More precisely, conditions (4.10) and (4.11) differ by a few (electron
volts)/c2 , associated with the electronic binding energy differences, and
since ÿ-decay usually takes place in an atomic environment (4.11) is the
more suitable form. The energy released in nuclear ÿ-decay is never large
enough to produce particles other than electrons or positrons, and neutrinos.
77
As an example, 77
32 Ge decays by a series of ÿ-decays to 34 Se, Z increasing by one at each stage:
77
32 Ge
ÿ
! 77
33 As ‡ e ‡ e ‡ 2:75 MeV
#
77
34 Se
‡ eÿ ‡ e ‡ 0:68 MeV:
77
34 Se
is the only stable nucleus with A ˆ 77.
A nucleus with Z > Zmin can decay by emitting a positron and a
neutrino. For example, another sequence of decays ending in 77
34 Se is:
77
36 Kr
‡
! 77
35 Br ‡ e ‡ e ‡ 2:89 MeV
#
77
34 Se
‡ e‡ ‡ e ‡ 1:36 MeV:
For the process of ÿ-decay by positron emission to be possible the condition is
mnuc A; Z† > mnuc A; Z ÿ 1† ‡ me ;
or, in terms of atomic masses,
ma A; Z† > ma A; Z ÿ 1† ‡ 2me :
4:12†
In an atomic environment, a ÿ-decay process competing with positron emission is electron capture, in which the nucleus absorbs one of its
cloud of atomic electrons, emitting only a neutrino. For example,
46
Nuclear sizes and nuclear masses
77
35 Br
‡ eÿ !77
34 Se ‡ e ‡ 2:38 MeV:
Such processes are often referred to as K-capture, since the electron is
most likely to come from the innermost `K-shell' of atomic electrons. The
condition for K-capture to be possible is less restrictive than (4.12):
mnuc A; Z† ‡ me > mnuc A; Z ÿ 1† ‡ be =c2 ;
where the main contribution to the electronic energy be is the binding
energy of the K electron, or
ma A; Z† > ma A; Z ÿ 1†;
4:13†
where ma is the mass of the excited atom. For example, 74 Be decays by Kcapture:
7
4 Be
‡ eÿ ! 73Li ‡ e ‡ 0:86 MeV;
whereas it cannot decay by positron emission. When both processes are
possible, the energy release in K-capture will be 2me c2 1 MeV greater
than in the corresponding positron emission.
The vacancy in the atomic K-shell will be ®lled by an electron falling
from a less bound atomic shell. The energy released in this transition will
appear either in the emission of a photon (X-ray), or in the ejection from
the atom of an Auger electron, usually from the L-shell. The latter process
results from the Coulomb interaction between the electrons.
Thus odd-A nuclei decay to the value of Z closest to ÿ=2þ. It is clearly
highly unlikely that there will be two values of Z giving exactly the same
atomic masses; we expect there to be only one ÿ-stable Z value for odd-A
nuclei, and such is the case.
Nuclei with even A must have Z and N both even numbers, or Z and
N both odd numbers. In the semi-empirical mass formula, the even±even
1
nuclei have a lower energy than the odd±odd nuclei by 2Aÿ2 . This quantity varies from 5 MeV when A ˆ 20 to 1.4 MeV when A ˆ 250. Thus
1
there are two mass parabolas with relative vertical displacement 2Aÿ2 =c2 ,
as in Fig. 4.5, for each even value of A.
In Fig. 4.5 the values Z ˆ 28 and Z ˆ 30 on the lower even±even
parabola may both be regarded as effectively stable with respect to ÿdecay. In this particular example the only energetically possible ÿ-decay
process linking the two would be the `double K-capture'
4.6 The ÿ-stability valley
47
Fig. 4.5 The atomic masses of atoms with A ˆ 64 relative to the atomic mass of
64
28 Ni. Open circles * are odd±odd nuclei, ®lled circles * are even±even nuclei.
The theoretical even±even and odd±odd parabolas are drawn using the paraÿ
meters of equation (4.5). Note the odd±odd nucleus 64
29 Cu, which can ÿ -decay
64
‡
64
to 30 Zn or ÿ -decay to 28 Ni, both of which are stable, naturally occurring, isotopes. These decays are discussed in detail in Chapter 12.
64
30 Zn
‡ 2eÿ !64
28 Ni ‡ 2e ‡ 1:1 MeV:
This decay has not been observed though it is theoretically possible.
Processes with the simultaneous emission of two electrons or two
positrons, or the simultaneous absorption of two electrons, have been
much investigated, both experimentally and theoretically. Experimentally, the ®rst direct laboratory observation of such a process was
made in 1987, with the double ÿ-decay
82
34 Se
ÿ
!82
36 Kr ‡ 2e ‡ 2 e ‡ 3:03 MeV:
The mean lifetime for this decay was measured to be 1:6 1020 yr.
Measurements of such long lifetimes are dif®cult (see Problem 4.10).
Several other double ÿ-decays have been observed since, all with lifetimes
48
Nuclear sizes and nuclear masses
similar in magnitude to that of 82 Se. These measurements are in good
agreement with the theoretical estimates which have been made within the
Standard Model (}3.7). Much of the interest in double ÿ-decay measurements stems from the possibility of observing other modes of ÿ-decay
which are predicted in certain extensions of the Standard Model.
Figure 4.5 is characteristic of nuclei with even A, and pairs of stable
nuclei with different (even) Z but the same A are common. The only odd±
odd nuclei which are stable are the four lightest: 21 H, 63 Li, 105 B and 147 N ±
but for A < 20 the semi-empirical mass formula is less accurate.
The nuclei which are observed to be ÿ-stable are plotted in Fig. 4.6 as
points in the N; Z† plane. Nuclei of constant A lie on the diagonal lines
N ‡ Z ˆ A. The bottom of the `ÿ-stability valley' where the ÿ-stable
nuclei are found is given remarkably well by the approximation (equation
(4.9))
Z ˆ ÿ=2þ ˆ
4.7
4s ‡ mn ÿ mp ÿ me †c2 †A
2
2 4s ‡ dA3 †
:
4:14†
The masses of the ÿ-stable nuclei
With the approximation Z ˆ ÿ=2þ, the binding energies of the ÿ-stable
nuclei can be calculated from equation (4.5). For odd-A nuclei the pairing
energy term is zero, and the resulting binding energy per nucleon B A†=A
is plotted against A in Fig. 4.7 and the various contributions to B A†=A
are displayed in Fig. 4.8.
It should be noted that apart from pairing effects the bulk term is the
only positive contribution to the binding energy. The initial rise of B=A
with A is simply due to the negative surface contribution diminishing in
magnitude relative to the bulk contribution as the size of the nucleus
increases. However, as A and therefore Z increase further, the
Coulomb term becomes important and produces a maximum on the
curve.
The curve gives the observed nuclear-binding energies quite well. The
small deviations of the experimental values from the smooth curve are for
the most part due to the quantum-mechanical `shell' effects, which are
considered in the next chapter. The maximum binding energies lie in the
neighbourhood of 56 Fe.
4.7 The masses of the ÿ-stable nuclei
49
Fig. 4.6 The ÿ-stability valley. Filled squares denote the stable nuclei and longlived nuclei occurring in nature. Neighbouring nuclei are unstable. Those for
which data on masses and mean lives are known ®ll the area bounded by the
lines. For the most part these unstable nuclei have been made arti®cially. (Data
taken from Chart of the Nuclides (1977), Schenectady: General Electric
Company.)
50
Nuclear sizes and nuclear masses
Fig. 4.7 The binding energy per nucleon of ÿ-stable (odd-A) nuclei. Note the
displaced origin. The smooth curve is from the semi-empirical mass formula with
Z related to A by equation (4.14). Experimental values for odd-A nuclei are
shown for comparison; the main deviations (<1%) are due to `shell' effects not
included in our formula.
4.8
The energetics of -decay and ®ssion
The peak in the binding energy curve makes possible other modes of
decay for a heavy nucleus which is stable against ÿ-decay. Since there is
a gradual decrease of B=A† with A for the heavier nuclei, it may be
energetically advantageous for a heavy nucleus to split into two smaller
nuclei, which together have a greater net binding energy. The most common such process is the emission of an -particle. As Table 4.2 shows,
4.8 The energetics of -decay and fission
51
Fig. 4.8 The contributions to B=A. Note that the surface, asymmetry and
Coulomb terms all subtract from the bulk term.
4
2 He
has the comparatively large binding energy of 28.3 MeV. The condition for -emission to be possible from a nucleus A; Z† to give a
nucleus A ÿ 4; Z ÿ 2† is
B A; Z† < B A ÿ 4; Z ÿ 2† ‡ 28:3 MeV:
4:15†
For A; Z† on the line of ÿ-stability, this condition is always satis®ed for
suf®ciently large A, A0165, and all such nuclei are, in principle, able to
emit -particles. However, we shall see in Chapter 6, where the physical
mechanism of -decay is analysed, that decay rates are so slow that the ÿ-
52
Nuclear sizes and nuclear masses
stable nuclei can also be regarded as -stable up to 209
83 Bi. Beyond, only
some isotopes of Th and U are suf®ciently long-lived to have survived on
Earth since its formation; other unstable heavy elements are produced
either from the decay of these, or arti®cially.
Another energetically favourable process which is possible when A is
large is the splitting of a nucleus into two more nearly equal parts. This is
called ®ssion. The energetics of ®ssion may be explored using the semiempirical mass formula, and in Chapter 6 we shall investigate the rate of
spontaneous ®ssion processes.
Beyond the heavy elements of the actinide group, -decay and ®ssion
bring the Periodic Table to an end.
4.9
Nuclear binding and the nucleon±nucleon potential
To what extent do the nuclear properties discussed in this chapter follow
from the nucleon±nucleon potential introduced in Chapter 3? Much theoretical effort has been expended on this question. In a nucleus containing three or more nucleons, the nuclear potential energy need not be the
simple sum of two-body potentials over all pairs of nucleons: since the
nucleons are composite particles, there may well be additional interactions.
Even if the possibility of additional interactions is not considered, the
computations are not easy but it appears that the two-body potentials are
the dominant contribution to the nuclear potential energy. For `bulk'
nuclear matter the Paris potential gives a value of 16 MeV/nucleon for
the binding energy per nucleon, in good agreement with values found for
the parameter a in the semi-empirical mass formula (4.5). However, the
calculated density of nuclear matter is somewhat too high. The Paris
potential gives 0.94 fm rather than the empirical 1.12 fm for the parameter in (4.2).
Similar semi-quantitative agreement is found when the two-nucleon
potential is applied to particular light nuclei. For example, the binding
energy of 31 H is calculated to be 7.38 MeV, and the experimental value
(Table 4.2) is 8.48 MeV.
Problems
4.1 A relativistic electron whose rest mass can be neglected has energy E. It
scatters elastically from a particle of mass M at rest and after the collision has turned through an angle and has energy E 0 .
Problems
53
(a) Show that the total energy of the struck particle after the collision is
EM ˆ E ÿ E 0 ‡ Mc2 .
(b) Show that its momentum is
1
PM ˆ ‰E 2 ‡ E 02 ÿ 2EE 0 cos Š2 =c.
2
ˆ P2M c2 ‡ M 2 c4 † show that the fraction of energy lost
(c) Hence (using EM
by the electron is
E ÿE0
1
:
ˆ
E
1 ‡ Mc2 =‰E 1 ÿ cos †Š
For E a few hundred MeV, show that this is small if the struck particle
is a heavy nucleus, and is large (except for 0) if the struck particle is
an electron.
4.2 In quantum mechanics, the differential cross-section for the elastic scattering of a relativistic electron with energy E 4 me c2 from a ®xed electrostatic potential c r† is given in the Born approximation, and
neglecting the effects of electron spin, by
2 4 Z
2
d
E
1
ˆ
e c r†e iqr d3 r
dÿ
2
»c
where q is the difference between the ®nal and the initial wave-vectors of
the electron.
(a) Show that q ˆ jqj ˆ 2E=»c† sin =2†, where is the scattering angle.
(b) Poisson's equation relates the potential c r† to the charge density ech r†
by r2 c ˆ ÿech ="0 . Noting r2 e iqr ˆ ÿq2 e iqr , and integrating by parts,
show that
!2
2 4 ÿ 2 Z
d
E
1 1 e
iqr 3
r†e
d
r
:
ˆ
ch
dÿ
2
»c q4 "0
For light nuclei (for which the Born approximation has a greater validity) a measured cross-section can be used to infer the Fourier transform
of the charge distribution, as this example indicates.
4.3 Show that the characteristic velocity v of a lepton of mass m bound in an
1
atomic orbit is given by v=c »=amc ˆ 137
, where a ˆ 4"0 † »2 =me2 is
the appropriate Bohr radius for that lepton. Hence show that the muon
mean life is long compared with the characteristic time scale a=v for its
motion in an atomic orbit.
4.4 The ground-state wave-function of a lepton of mass m in a Coulomb
potential ÿZe2 = 4"0 r† is
54
Nuclear sizes and nuclear masses
3
1 Z 2 ÿZr=a
ý r† ˆ p
e
a
where a ˆ 4"0 † »2 =me2 , and the corresponding binding energy E is
Z 2 »2 =2ma2 .
The ®nite size of the nucleus modi®es the Coulomb energy for r < R,
the nuclear radius, by adding a term of the approximate form
"
#
Ze2 3
r2
R
V r† ˆ ÿ
ÿ
:
ÿ
4"0 R 2 2R2 r
(a) Show that the volume integral of this potential is
Z
V r†d3 r ˆ
Ze2 R2
:
10"0
(b) Show that the ®rst-order correction to the binding energy due to this
R
term, E ˆ ý r†V r†ý r†d3 r, is
E e2 Z4 R2
:
10"0 a3
(Note that the lepton wave-function can be taken to be constant over
nuclear dimensions.)
(c) For the nucleus
E
5 10ÿ6
E
66
30 Zn
show that
for electrons,
E
0:2 for muons.
E
4.5 Using Table 4.2 show that 84 Be can decay to two -particles with an
energy release of 0.1 MeV, but that 126 C cannot decay to three -particles. Show that the energy released (including the energy of the photon)
in the reaction 21 H ‡ 42 He ! 63 Li ‡ þ is 1.5 MeV.
4.6 Consider nuclei with small nucleon number A and such that
Z ˆ N ˆ A=2. Neglecting the pairing term, show that the semi-empirical
mass formula then gives the binding energy per nucleon
1
2
B=A ˆ a ÿ bAÿ3 ÿ d=4†A3 :
Show that this expression reaches a maximum for Z ˆ A=2 ˆ 26 (iron).
4.7 Using the formula (4.14) calculate Z for A ˆ 100 and A ˆ 200. Compare
your results with Fig. 4.6 and comment.
Problems
55
4.8 The carbon isotope 146 C is produced in nuclear reactions of cosmic rays
in the atmosphere. It is ÿ-unstable.
14
6C
! 147 N ‡ eÿ ‡ e ‡ 0:156 MeV,
with a mean life of 8270 years.
It is found that a gram of carbon, newly extracted from the atmosphere, gives on average 15.3 such radioactive decays per minute. What
is the proportion of 14 C isotope in the carbon?
What count rate would you expect from one gram of carbon extracted
from the remains of a wooden hut thought to be 4000 years old?
4.9 In an experiment using 14 g of selenium containing 97% by weight of
82
34 Se, 35 events associated with the double ÿ-decay
82
34 Se
ÿ
! 82
36 Kr ‡ 2e ‡ 2e
were counted over a period of 7960 hours. Assuming a detector ef®ciency of 6.2%, estimate the mean life for this decay.
(See Elliott, S. R., Haln, A. A. and Moe, M. K. (1987), Phys. Rev. Lett.
59, 2020.)
4.10 If a double ÿ-decay without the emission of neutrinos were possible,
what would be its experimental signature?
4.11 On the basis of the different properties of nuclei with even A and with
odd A, explain why there are about 300 ÿ-stable nuclei with masses up to
that of 209
83 Bi. What is the average number of isotopes per element?
5
Ground-state properties of nuclei:
the shell model
5.1
Nuclear potential wells
In the last chapter, we set out a semi-empirical theory for the binding
energy of an atomic nucleus, and quantum-mechanical considerations
came in only rather indirectly. Experimental atomic masses show deviations from the smooth curve given by the semi-empirical mass formula,
deviations which we said were of quantum-mechanical origin. Since a
nucleus in its ground state is a quantum system of ®nite size, it has
angular momentum J, with quantum number j which is some integral
multiple of 12. If j 6ˆ 0 the nucleus will have a magnetic dipole moment,
and it may have an electric quadrupole moment as well.
The nuclear angular momentum and magnetic moment manifest
themselves most immediately in atomic spectroscopy, where the interaction between the nuclear magnetic moment and the electron magnetic
moments gives rise to the hyper®ne structures of the electronic energy
levels. In favourable cases both j and the magnetic moment may be
deduced from this hyper®ne splitting.
The observed values of nuclear angular momenta give strong support
to the validity of a simple quantum-mechanical model of the nucleus: the
nuclear shell model. In this model, each neutron moves independently in a
common potential well that is the spherical average of the nuclear potential produced by all the other nucleons, and each proton moves independently in a common potential well that is the spherical average of the
nuclear potential of all the other nucleons, together with the Coulomb
potential of the other protons. Since the nuclear forces are of short range
we can guess that the shape of such an average nuclear potential will
56
5.1 Nuclear potential wells
57
re¯ect the nuclear density exhibited in Fig. 4.4. In particular it will be
uniform in the central region and rise steeply in the surface region.
Indeed, we can anticipate that the steep rise will be even more pronounced than the fall in nuclear density, since tunnelling into the classically forbidden region will make the nuclear density near the surface more
diffuse. Thus the potential wells for neutrons and protons will be, qualitatively, as sketched in Fig. 5.1(a) and Fig. 5.1(b).
In order to model the effect of proton charge, we have added to the
proton well the Coulomb potential energy Uc r† of a proton in a sphere of
uniform charge density and total charge Z ÿ 1†e, corresponding to a
uniform distribution of the other Z ÿ 1† protons. (Such a charge distribution gave an energy which agreed well with the empirical Coulomb
contribution in the mass formula.) Elementary electrostatics gives
"
#
8
>
Z ÿ 1†e2 3
r2
>
>
;
ÿ
<
4"0 R 2 2R2
Uc r† ˆ
>
>
Z ÿ 1†e2
>
:
;
4"0 r
r < R;
5:1†
r > R;
where R is the nuclear radius.
Since the basic nucleon±nucleon interaction is state-dependent, there
are other factors which affect the relative depths of the neutron and
proton wells. For nuclei with more neutrons than protons the contribution of the strong nucleon±nucleon interaction to the potential is more
Fig. 5.1 A schematic representation of (a) the neutron potential well and (b)
F
F
the proton potential well for the nucleus 208
82 Pb. En and EP ÿ U† have been
estimated using equation (5.5). The observed neutron separation energy Sn of
7.4 MeV implies a neutron well-depth of 51 MeV. The observed proton separation energy Sp of 8.9 MeV implies that U ˆ 11 MeV. U represents the sum of the
mean electrostatic potential and the asymmetry energy.
58
Ground-state properties of nuclei: the shell model
attractive for the protons than for the neutrons, since a proton in such a
nucleus is, on average, more subject to the neutron±proton interaction
(recall the discussion of symmetry energy in }4.5). This lowers the proton
well-depth relative to Uc r†, as is indicated in Fig. 5.1(b) (which is drawn
with parameters appropriate to 208
82 Pb). The nuclear shell model treats all
these effects empirically.
5.2
Estimates of nucleon energies
Let us, for the moment, disregard the details of the nuclear potential at
the surface, and replace it by a potential with in®nitely high walls at
r ˆ R, which force the nucleon wave-functions to be zero at, and outside,
r ˆ R. We measure energies from the bottom of the neutron well, and for
simplicity take the proton well to be raised with respect to the neutron
This U represents the mean electrostatic
well by a constant energy U.
potential and any asymmetry contributions to the proton potential well.
Then the SchroÈdinger equations for the neutron states þn and proton
states þp are respectively
ÿ
»2 2
r þn ˆ En þn ;
2mn
5:2†
ÿ
»2 2
p;
r þp ˆ Ep ÿ U†þ
2mp
5:3†
where any terms involving the intrinsic spin of the nucleons have been
neglected.
Nucleons are fermions, and the Pauli exclusion principle requires that
no two neutrons nor two protons are in the same state. Hence in the shell
model of the ground state of a nucleus with N neutrons and Z protons,
the lowest N neutron states are occupied up to some energy EnF , called the
neutron Fermi energy, and the lowest Z proton states are occupied up to
some energy EpF , the proton Fermi energy. To obtain a qualitative estimate of the energies involved, we suppose that N and Z are suf®ciently
large for us to use the elementary formula for the integrated density of
states N E†, derived in Appendix B,
3
V 2mE 2
N E† ˆ 2
;
3
»2
5:4†
5.2 Estimates of nucleon energies
59
where V is the volume of the system considered, in this case the nuclear
volume V ˆ 4R3 =3. Hence EnF and EpF are given by
ÿ
!32
V 2mn EnF
N 2
;
3
»2
ÿ
!3
F
2
V 2mp Ep ÿ U†
Z 2
;
3
»2
5:5†
since proton kinetic energies are given by Ep ÿ U†.
In lighter nuclei, A940, the number of neutrons and protons is
approximately equal, so that N=V is close to half the `density of nuclear
matter' given by equation (4.2), i.e. N=V 0:085 fmÿ3 . For this density,
equation (5.5) gives EnF 38 MeV, irrespective of the particular nucleus.
For heavier nuclei, this ®gure will increase somewhat, but for energies of
this order the corresponding neutron velocities are quite low: v2F =c2 0:1.
This gives some justi®cation for our use of the non-relativistic
SchroÈdinger equation for the neutrons. Similarly, a proton at the Fermi
38 MeV also.
energy EpF has kinetic energy EpF ÿ U†
Similar energies are to be expected in more realistic potentials of ®nite
depth. In a ®nite well, the depth of EnF below the external potential outside the nucleus is equal to the energy required to detach a neutron from
the nucleus. This energy, the neutron separation energy Sn , is given in
terms of binding energies by
Sn N; Z† ˆ B N; Z† ÿ B N ÿ 1; Z†;
5:6†
and hence is of the order of the binding energy per nucleon, about 8 MeV
(Fig. 4.7, see also Table 4.2). Thus the total depth of the neutron well is
46 MeV.
Similarly the proton separation energy is de®ned by
Sp N; Z† ˆ B N; Z† ÿ B N; Z ÿ 1†:
The most stable nucleus of a given mass number A ˆ N ‡ Z will have
the neutron and proton Fermi energies approximately equal: if they were
to differ in energy by more than (me c2 ‡ energy level spacing), the
nucleus would be unstable to ÿ-decay. A nucleon at the higher Fermi
energy would decay to an empty state just above the lower Fermi energy,
to form a new nucleus with different charge, and lower total energy. An
equivalent condition is that Sn and Sp must be approximately equal.
The characteristic velocity of a nucleon at the Fermi energy, and the
radius R of the nucleus, set a typical nuclear time scale tnuc :
60
Ground-state properties of nuclei: the shell model
tnuc ˆ
5.3
1
2R
2:6 10ÿ23 A3 s:
vF
5:7†
Energy shells and angular momentum
To obtain more precise information about the nucleon levels we must
solve the SchroÈdinger equations (5.2) and (5.3). For simplicity, we again
take the potential wells with in®nite walls. The SchroÈdinger equations are
separable in r; ; † coordinates so that
þ r; ; † ˆ ul r†Ylm ; †;
where l ˆ 0; 1; 2; 3; . . . and m ˆ ÿl; ÿl ‡ 1; . . . ; l ÿ 1; l, are orbital angular momentum quantum numbers (Appendix C). States with l ˆ 0; 1; 2;
3; 4; 5; . . . are called s, p, d, f, g, h, . . . states, the notation having been
established in the early days of atomic spectroscopy. Taking equation
(5.2) for neutrons (and the equation for protons differs only in the shift
U in energy), the radial function ul r† satis®es
ÿ»2 1 d2
» l l ‡ 1†
rul † ‡
ul ˆ Eul ;
2
2mn r dr
2mn r2
5:8†
with the boundary conditions that u r† is ®nite at r ˆ 0 and zero at r ˆ R.
When l ˆ 0 (s-states) we see immediately that the solutions ®nite at
r ˆ 0 are
us r† ˆ
sin kr†
kr
with E ˆ
» 2 k2
:
2mn
5:9†
The boundary condition at r ˆ R is satis®ed by taking k ˆ kn ˆ n=R,
where n ˆ 1; 2; 3; . . ., and the corresponding energy levels 1s, 2s, 3s, . . .,
are given by
E n; s† ˆ
»2 n2
:
2mn R
5:10†
Thus there is a sequence of energy eigenstates having l ˆ 0, labelled by
the additional quantum number n. When l ˆ 1 (p-states) the SchroÈdinger
equation is
5.3 Energy shells and angular momentum
ÿ
»2 1 d2
»2
rup † ‡
up ˆ Eup ;
2
2mn r dr
mn r2
61
5:11†
and it is straightforward to check by differentiation that the solution
®nite at r ˆ 0 is
up r† ˆ
sin kr† cos kr†
ÿ
;
kr†
kr†2
Eˆ
»2 2
k :
2mn
5:12†
We must again choose k so that up R† ˆ 0. Let us write kR ˆ x. The
values of x for which up R† ˆ 0 are x1p ˆ 4:49, x2p ˆ 7:73; . . ., and the
corresponding energies are E n; p† ˆ »2 =2mn † xnp =R†2 . It should be
noted that the labelling of energy levels, n ˆ 1; 2; 3; . . ., for each l
value, differs from that conventionally adopted in atomic physics.
In fact us r† and up r† are special cases of the spherical Bessel functions
jl kr†. For arbitrary l the zeros of jl x†, which give the allowed values of k,
are tabulated in standard tables of mathematical functions, and given in
column (2) of Table 5.1. Thus for any l the levels E n; l† ˆ »2 =2m† xn1 =
R†2 are easily determined, and hence their sequence in order of increasing
energy. This is given in columns (1) and (7) of Table 5.1. For each E n; l†,
there are 2l ‡ 1† allowed values of the quantum number m. Since we have
so far neglected any coupling between the intrinsic spin of a nucleon and
its orbital motion, each nucleon has two possible spin states, which may
be characterised by ms ˆ 12 ; ms ˆ ÿ 12, so that there are 4l ‡ 2† states of
the same energy for a given n; l†.
The sequence of the levels E n; l† is not very sensitive to the precise
details of the well, and is much the same for a well of ®nite depth and
appropriately rounded shape. If N neutrons are put into the neutron well,
the ground state (which may be degenerate) will correspond to the occupation of the N lowest lying energy states. Figure 5.2 expresses graphically the number of states available in terms of the dimensionless quantity
x ˆ kR. N x† is the number of states with energy less than »2 =2mn †
x=R†2 or, equivalently, the number of zeros xnl with xnl < x, each zero
being counted 4l ‡ 2† times. Also drawn is the asymptotic formula, valid
for large x,
2
ÿ
!12 3
3
4x3
9
V 2mn E 2 4
3 S
»2
5:
1ÿ
N x† ˆ
ˆ 2
1ÿ
9
8x
8 V 2mn E
3
»2
5:13†
62
Ground-state properties of nuclei: the shell model
Table 5.1
xnl
Neutron
Proton
1s
3.14
1s1
2
2
1s12
1s
1p
4.49
1p32
1p12
6
8
6
8
1p32
1p12
1p
1d
5.76
1d52
2s12
1d
6.28
14
16
20
14
16
2s
1d52
2s12
1d32
20
1d32
1f 72
28
2p32
32
28
1f 72
1f 52
2p12
1g92
38
40
50
32
38
2p32
1f 52
2d52
1g72
1h112
56
64
76
40
2p12
50
1g92
3s12
2d32
78
82
58
1g72
2f 72
1h92
3p32
1i132
2f 52
3p12
90
100
104
118
124
126
64
2d52
76
80
82
1h112
2d32
3s12
2g92
1i112
2g72
136
148
156
1f
2p
6.99
7.73
1g
8.18
2d
9.10
1h
9.36
3s
9.42
2f
1i
10.42
10.51
3p
10.90
1j
11.66
2g
11.70
2
2s
1f
2p
1g
2d
92
100
1h
3s
1h92
2f 72
The ®rst and last columns give the sequence of energy levels in a spherical well with in®nite
walls. The second column gives the corresponding values of xnl ˆ knl R. The third column
gives the observed sequence of spin±orbit coupled levels for neutrons, and the fourth the
cumulative number of available states in these levels. The remaining two columns give the
levels and number of states for protons. The spacings are chosen so that the ®lling of the
neutron and proton shells for stable nuclei is approximately in step down the columns. Lines
are drawn at the `magic numbers'.
5.3 Energy shells and angular momentum
63
Fig. 5.2 The exact N x† compared with the asymptotic formula of equation
(5.13). (See Baltes, H. P. and Hilf, E. R. (1976), Spectra of Finite Systems, for
the derivation of (5.13).)
This is an extension of the usual density of states formula we used in
equation (5.4) and includes a correction for the effects of the surface (of
area S). It follows the exact N x† remarkably closely.
So far we have neglected spin. The crucial step in establishing the
nuclear shell model was the recognition that there must also be a spin±
orbit coupling term in the self-consistent potential seen by the nucleons, of
the form
Uso r†L s:
5:14†
A term like this is not perhaps too surprising since in the basic nucleon±
nucleon interaction, equation (3.4), there is coupling between the spins
and the orbital motions of the nucleons.
With the introduction of this spin±orbit coupling term into the potential, L2 and s2 are still conserved since
‰L2 ; L sŠ ˆ 0;
‰s2 ; L sŠ ˆ 0;
5:15†
64
Ground-state properties of nuclei: the shell model
so that l and s ˆ 12† remain good quantum numbers, but m and ms , are
`good quantum numbers' no longer, since
‰Lz ; L sŠ 6ˆ 0;
‰ms ; L sŠ 6ˆ 0:
5:16†
Thus the magnitudes, but not the directions, of L and s are preserved.
However, the total angular momentum J ˆ L ‡ s is of course conserved,
so that states may be speci®ed by the quantum numbers l; s; j; jz †. In
Appendix C it is shown that for a given l and s ˆ 12, the allowed values
of j are: l ‡ 12, with ‰2 l ‡ 12† ‡ 1Š ˆ 2l ‡ 2 allowed values of jz ; and l ÿ 12,
with ‰2 l ÿ 12† ‡ 1Š ˆ 2l allowed values of jz . The parity of the state speci®ed by l; s; l 12 ; jz † is ÿ1†l .
The expectation value of L s may be obtained from the identity
L s ˆ 12 ‰ L ‡ s†2 ÿ L2 ÿ s2 Š ˆ 12 ‰J2 ÿ L2 ÿ s2 Š;
5:17†
so that
l; s; j; jz jL sjl; s; j; jz † ˆ 12‰ j j ‡ 1† ÿ l l ‡ 1† ÿ s s ‡ 1†Š »2
(
1 2
l»
if j ˆ l ‡ 12
ˆ 2
ÿ12 l ‡ 1† »2 if j ˆ 1 ÿ 12:
5:18†
Thus the introduction of spin±orbit coupling splits the 4l ‡ 2†-fold
degenerate level n; l† into two levels which we may label by nll‡12 , nllÿ12 .
For example, when l ˆ 2 (d-states),
nd (10 states) ! nd52 (6 states) and nd32 (4 states):
Experiment shows that the sign of Uso r† is negative so that the state
with j ˆ l ‡ 12 always has lower energy than the state with j ˆ l ÿ 12.
Equation (5.18) suggests that the energy splitting increases with l, though
of course the form of the radial function is also relevant in the calculation
of the energy levels. The splitting is great enough to change the orbital
sequence of columns (1) and (7) of Table 5.1. This effect is most apparent
in the heavier nuclei, where because of increasing R the orbital levels are
closer together in energy than in the lighter nuclei.
The sequence of `shells' inferred from experiment is shown in columns
(3) and (6) for neutrons and protons respectively. The shift of the proton
5.4 Magic numbers
65
column relative to the neutron column re¯ects the shell ®lling observed
for the ÿ-stable nuclei. In the ®lling of these levels there are departures
from the order given in the case of a few particular nuclei.
The major success of the shell model is the prediction of the angular
momenta of nuclei in their ground states. These values follow simple
rules: nuclei with an even number of protons and an even number of
neutrons (even±even nuclei) have angular momentum zero and positive
parity, nuclei with an even number of protons and an odd number of
neutrons or vice-versa (even±odd nuclei), have angular momentum and
parity equal to that of the odd nucleon in the shell that is being ®lled. We
saw in Chapter 4 that it was energetically favourable for nuclei to contain
even numbers of protons and even numbers of neutrons. The information
from nuclear spins makes more precise the origin of this energy: it seems
that it is energetically advantageous for nuclei to take pairs of protons
and pairs of neutrons into the energy shells, with the angular momenta of
the pairs coupled to zero, J1 ‡ J2 ˆ 0, so that the angular momentum
and parity of an unpaired nucleon is the angular momentum and parity of
the whole nucleus. There are some exceptions to this last rule, but
remarkably few considering its simplicity.
In the case of odd±odd nuclei, the odd proton and odd neutron do
not combine their angular momenta in any systematic way; there is no
very clear empirical rule, and no simple theory. Indeed, odd±odd nuclei
are altogether energetically disfavoured. There are only four stable odd±
odd nuclei 21 H; 63 Li;105 B;147 N†, the rest undergo ÿ-decay to become even±
even (Chapter 4).
The rules may be seen obeyed by the light nuclei of Table 4.2. For
‡
example 178 O has one odd neutron in the 1d52 shell, and spin and parity 52 .
5.4
Magic numbers
In Table 5.1 lines have been drawn where the total numbers of states in
the shells above the line are 2, 8, 20, 28, 50, 82 and 126, the so-called
`magic numbers'. The ®rst two numbers just correspond to the ®lling of 1s
and 1p shells. The others appear to be somewhat arbitrary, but it is found
empirically that the energy gaps to the next shell are greater than average
at these points in the sequence. Nuclei having Z or N equal to one of
these numbers have properties which re¯ect the existence of such a gap.
For example, tin Z ˆ 50† has ten stable isotopes, and there are seven
stable elements having N ˆ 82 (see Fig. 4.6). These examples illustrate
that there must be a large gap in the energy spacing at Z ˆ 50 in the
66
Ground-state properties of nuclei: the shell model
proton sequence and at N ˆ 82 in the neutron sequence, as is clear from
our discussion in }5.2 of the condition EnF EpF .
The departures from the smooth curve of binding energy per nucleon,
given by the semi-empirical mass formula, are associated with the nuclear
shell structure and the magic numbers. Magic number nuclei are particularly strongly bound and have been marked on Fig. 4.7. The heaviest stable nuclei are 208
82 Pb, which is `doubly magic' with N ˆ 126, Z ˆ 82,
and 209
Bi
which
has
N ˆ 126.
83
The magic numbers were identi®ed experimentally before the shell
model was established, and indeed the indications of shell structure provided by the existence of these numbers was a strong motivation for the
formulation of the shell model. Other consequences of the magic numbers
will be mentioned in the sections on the excited states of nuclei and
atomic abundances in the Solar System.
5.5
The magnetic dipole moment of the nucleus
The successful description of nuclear angular momentum indicates the
essential validity of the shell model. The model also gives a qualitative
understanding of the magnetic dipole moments of nuclei. The magnetic
moments of paired nucleons, like their spins, cancel exactly, and all even±
even nuclei are found to have zero magnetic moments. A nucleus with
angular momentum operator J (quantum numbers j; jz ) has a magnetic
moment operator l which, averaged over the nucleons, must be aligned
with J, since J is the only vector available giving a preferred direction.
The magnetic dipole moment l is de®ned by writing
hli ˆ
hJi;
jȠ
5:19†
where the brackets h i indicate any matrix element between the 2j ‡ 1†
states labelled by jz .
In a magnetic ®eld B ˆ 0; 0; B†, which speci®es the z-direction, the
magnetic potential energy of the nucleus in the ®eld is the expectation
value of ÿl B ˆ ÿz B. For a state of given jz , this energy is, from
equation (5.19),
E jz † ˆ ÿ jz =j†B
5:20†
5.6 Calculation of the magnetic dipole moment
67
so that there are 2j ‡ 1† equally spaced energy levels corresponding to
jz ˆ ÿj; ÿj ‡ 1; . . . ; j.
Transitions between these levels may be induced by a radio-frequency
oscillating electromagnetic ®eld of angular frequency ! where
»! ˆ jjB=j:
5:21†
Measurements of this resonance frequency in a known magnetic ®eld give
a precise value for the magnetic dipole moment. The phenomenon is
called nuclear magnetic resonance, and has many applications in physics,
chemistry and biology.
5.6
Calculation of the magnetic dipole moment
In our simple version of the shell model, the magnetic moment of an oddA nucleus will arise entirely from the unpaired nucleon. If this unpaired
nucleon is a proton, its orbital motion will give, as in classical magnetism,
a moment
eL
L
lL ˆ
ˆ N
;
2mp
»
5:22†
where N ˆ e»=2mp is the nuclear magneton. A neutron, since it is
uncharged, will give no orbital contribution.
To this must be added the intrinsic magnetic moment of the nucleon,
ls ˆ gs N
s
;
»
5:23†
where (using the values quoted in equation (3.2)), gs ˆ 5:59 for a proton
and gs ˆ ÿ3:83 for a neutron.
Thus the total magnetic moment operator for a single nucleon is
l ˆ lL ‡ ls ˆ N ‰gL L ‡ gs sŠ= »;
5:24†
where gL ˆ 1 for a proton and gL ˆ 0 for a neutron. We can write this as
l ˆ lN ‰12 gL ‡ gs † L ‡ s† ‡ 12 gL ÿ gs † L ÿ s†Š= »;
and take the scalar product with J ˆ L ‡ s to give
68
Ground-state properties of nuclei: the shell model
l J ˆ N ‰12 gL ‡ gs †J2 ‡ 12 gL ÿ gs † L2 ÿ s2 †Š= »
(since L and s commute). Then the expectation value of each side of this
equation for a state speci®ed by j; jz ; l; s† gives, using equation (5.19)
j j ‡ 1† ˆ N ‰12 gL ‡ gs † j j ‡ 1† ‡ 12 gL ÿ gs † l l ‡ 1† ÿ s s ‡ 1††Š
j
so that
ˆ N
1
1
l ÿ s† l ‡ s ‡ 1†
:
gL ‡ gs † j ‡ gL ÿ gs †
2
2
j ‡ 1†
5:25†
Since s ˆ 12 and j ˆ l 12 we ®nally obtain for the contribution from the
upaired nucleon
for j ˆ l ‡ 12;
ˆ N ‰ jgL ÿ 12 gL ÿ gs †Š
j
ˆ N jgL ‡
g ÿ gs †
for j ˆ l ÿ 12;
2 j ‡ 1† L
5:26†
which are referred to as the `Schmidt values'.
These predictions of the simple model for a nucleus with an odd
unpaired nucleon are not grossly wrong: almost all the observed magnetic
moments for such nuclei lie between these two values. But they are not
accurate predictions and there is no generally accepted explanation of the
discrepancies. One possible reason is that the intrinsic magnetic moment
of the nucleon is smaller in a nuclear environment than in free space.
Another interpretation is that the magnetic moments provide a more
sensitive test of the nuclear shell model than does the nuclear spin, and
cooperative effects which we have neglected may contribute.
5.7
The electric quadrupole moment of the nucleus
Nuclei with spin 51 usually have small permanent electric quadrupole
moments. The size of this electric quadrupole moment gives an indication
of the extent to which the distribution of charge (and hence matter) in the
nucleus deviates from spherical symmetry. A nucleus is coupled through
its electric quadrupole moment to the gradient, at the nuclear site, of the
external electric ®eld produced by the molecular or crystalline environment of the nucleus. Like the nuclear magnetic dipole moment, the
5.7 The electric quadrupole moment of the nucleus
69
nuclear electric quadrupole moment provides a sensitive probe of this
environment for chemistry and condensed matter physics.
Classically, the energy of a nuclear charge distribution ech r† in an
external electrostatic potential r† is
Z
Uˆe
ch r† r†d3 r:
We take the origin r ˆ 0 to be the centre of mass of the nucleus. Since
ch r† is con®ned to the small nuclear volume we can approximate r† by
the ®rst few terms of a Taylor series,
r† 0† ÿ r E ‡
1X
xx ;
2 i;j i j ij
5:27†
where E ˆ ÿr is the electric ®eld, and
ij ˆ
@2 @E
ˆÿ i;
@xj
@xi @xj
all evaluated at r ˆ 0:
Here the indices i, j run from one to three, and we are using the notation
r ˆ x1 ; x2 ; x3 †, etc.
We then have
U ˆ eZ 0† ÿ E d ‡ 12e
X
i;j
Z
ij
ch r†xi xj d3 r;
R
where d ˆ e ch r†r d3 r is the electric dipole moment. The ®rst term
would be the energy if the nuclear charge Ze were a point charge at the
origin. The second term is the electric dipole energy. Apart from negligible weak interaction effects, nuclear charge densities have the re¯ection
symmetry ch r† ˆ ch ÿr†; thus nuclear electric dipole moments are zero,
and the effects of the extended nuclear charge distribution appear in the
term
U ˆ 12e
X
i;j
Z
ij
ch r†xi xj d3 r;
which in general does not vanish.
70
Ground-state properties of nuclei: the shell model
If we neglect the charge density of atomic electrons at the nucleus, the
external potential satis®es Laplace's equation
r2 ˆ
X
i
ij ˆ 0:
5:28†
We can therefore re-write U in the form
U ˆ
eX
Q ;
6 i;j ij ij
where
Z
Qij ˆ
ch r†‰3xi xj ÿ r2 ij Šd3 r:
Qij is de®ned to be the quadrupole moment tensor of the classical
charge distribution. ij is the usual Kronecker . The additional term
containing ij does not change U, because of (5.28), but makes Qij 0 if ch r† is spherically symmetric.
Dimensional analysis suggests that for a nucleus which has a nonvanishing quadrupole moment
jQij j=e nuclear dimension†2 :
A nucleus in a neutral atom is subject to the electric ®eld of the
surrounding atomic electrons. The closed shells of inner atomic electrons
will be distributed with near spherical symmetry, and will not contribute
appreciably to a quadrupole ®eld at the nucleus. Any quadrupole ®eld
will be produced by outer electrons, and it may therefore be anticipated
that
ij e= 4"0 † atomic dimension†3 :
Hence typically the interaction energy is of magnitude
U e2 nuclear dimension†2
10ÿ9 eV:
4"0 atomic dimension†3
5.7 The electric quadrupole moment of the nucleus
71
Such small energy shifts are detectable in radio-frequency spectroscopy, but it is clear that higher multi-pole moments arising from further
terms in the Taylor expansion (5.27) will be unimportant.
The charge distribution in a nucleus must of course be treated quantum mechanically rather than classically, and we de®ne the electric quadrupole moment operator by
Qij ˆ
X
‰3xpi xpj ÿ ij r2p Š;
protons p
where now the xpi are the proton coordinates.
Just as the matrix elements of the (vector) magnetic dipole operator l
are proportional to the matrix elements of the (vector) angular momentum operator J, it can be shown that for the tensor operator Qij
hQij i ˆ Ch‰32 Ji Jj ‡ Jj Ji † ÿ ij J2 Ši;
where C is a constant, and again the brackets indicate any matrix elements between the 2j ‡ 1† nuclear states of angular momentum quantum
number j, labelled by jz . Thus all the matrix elements of Qij are determined by a single quantity. It is conventional to take the expectation
value of Q33 in the state with jz equal to its maximum value j, and de®ne
this as the nuclear electric quadrupole moment Q, so that
Q ˆ C‰3j 2 ÿ j j ‡ 1†Š ˆ Cj 2j ÿ 1†;
or
Cˆ
Q
:
j 2j ÿ 1†
All other matrix elements are then determined in terms of Q. Note that Q
vanishes for nuclei with j ˆ 0 or j ˆ 12.
Experimental values of Q are obtained from spectral measurements
on systems in which the ®eld gradients can be accurately calculated. These
values are often very much larger than, and sometimes differ in sign from,
the predictions of the simple shell model. The implication is that the
deformation of many nuclei from spherical symmetry is much larger
than would be expected from the simple independent particle shell
model. In reality the deformations must result from collective effects
involving several nucleons. As might be expected, deviations from sphe-
72
Ground-state properties of nuclei: the shell model
rical symmetry are least in the neighbourhood of closed shells and largest
for nuclei with shells which are around half-full.
Problems
5.1 In the model of }5.2, verify that when
EnF ˆ EpF ÿ U 38 MeV.
N ˆ Z ˆ A=2 then
5.2(a) Show that in the model of }5.2 the total kinetic energy of a nucleus
containing N neutrons and Z protons is
‰35 NEnF ‡ 35 Z EpF ÿ U†Š:
(b) For N ÿ Z† 5 A, this expression may be Taylor expanded about
N0 ˆ A=2, Z0 ˆ A=2, E0F 38 MeV. Show that
"
#
2 N 1 N 2
F
F
;
E n E0 1 ‡
ÿ
3 N0
9 N0
"
2 #
ˆ E0F 1 ÿ 2 N ÿ 1 N
EpF ÿ U†
;
3 N0
9 N0
where N ˆ ÿZ ˆ N ÿ Z†=2.
(c) Hence show that the total kinetic energy of the nucleons in the nucleus is
approximately
3 F
5 E0 A
‡ 13 E0F
N ÿ Z†2
A
and therefore contributes ÿ23 MeV to the coef®cient `a' in the semiempirical mass formula, and 13 MeV to the symmetry coef®cient `s'
(equation (4.5)).
5.3(a) Show from equation (5.1) that the average Coulomb energy of a proton
in a nucleus of atomic number Z is
6 Z ÿ 1†e2
:
U c ˆ
5 4"0 R
(b) Show that for
5.4
208
82 Pb,
U c 2U.
40
20 Ca
is the heaviest stable nucleus with Z ˆ N. (It is doubly magic.) The
neutron separation energy is 15.6 MeV. Estimate the proton separation
energy, and compare your estimate with the empirical value of 8.3 MeV.
5.5 Suggest values for the spins and parities of the following nuclei in their
ground states:
31
15 P;
67
30 Zn;
115
49 In.
Problems
73
5.6 The measured spins, parities and magnetic moments of some nuclei are:
ÿ
‡
‡
7
9
3
43
93
137
; ÿ1:32N ; 41 Nb
; 6:17N ; 56 Ba
; 0:931N ;
20 Ca
2
2
2
‡
3
197
‡
; 0:145N ; 26
79 Au
13 Al 5 ; not known†:
2
Compare these values with the predictions of the shell model.
5.7 Calculate the nuclear magnetic resonance frequencies for (a) protons,
4
(b) 43
20 Ca (see Problem 5.6), in a magnetic ®eld of 1 tesla (= 10 gauss).
5.8 Show that for a uniformly charged ellipsoid of revolution
x2 ‡ y2 z2
‡ 2 4 1,
a2
b
of total charge Ze,
Qzz ˆ 25 Z b2 ÿ a2 †:
The nucleus 176
71 Lu has j ˆ 7 and a very large electric quadrupole moment
of 8.0 barns. Suppose the nucleus in the state with jz ˆ 7 has approximately an ellipsoidal charge distribution of the form above. Calculate a
and b.
6
Alpha decay and spontaneous ®ssion
6.1
Energy release in -decay
We saw at the end of Chapter 4 that the binding energy per nucleon curve
of the ÿ-stable nuclei has a maximum in the neighbourhood of iron (Fig.
4.7), and that the heavier elements may be unstable to spontaneous disintegration. The principal mode of break-up is by emission of a 42 He
nucleus. Historically, the particles emitted in the decays of naturally
occurring -unstable nuclei were called -particles before they were identi®ed by Rutherford in 1908 as 42 He nuclei, and the name has stayed.
The kinetic energy release Q A; Z† in an -decay of a nucleus A; Z† is
given in terms of the binding energies of the parent and daughter nuclei
by
Q A; Z† ˆ B A ÿ 4; Z ÿ 2† ‡ 28:3 MeV ÿ B A; Z†;
6:1†
(where 28.3 MeV is the experimental binding energy of the 42 He nucleus). If
the nucleus is assumed to lie on the ÿ-stability curve, given approximately
by equation (4.14), then Q may be calculated as a function of Z (or of A)
from the semi-empirical mass formula, using equation (4.5). Neglecting
the pairing energy term, the effects of which are small, the resulting
smooth Q Z† is plotted in Fig. 6.1 for Z > 50. Negative values of Q
imply absolute stability against -decay. Also shown are Q values calculated from the experimentally measured masses of ÿ-stable nuclei and their
corresponding daughter nuclei. The trend of the experimental points is
given correctly by the semi-empirical formula. Though the detailed predictions can be out by as much as 5 MeV, this is very small (<0.3%)
74
6.2 The theory of -decay
75
Fig. 6.1 Experimental and theoretical -decay energies Q ˆ B A ÿ 4; Z ÿ 2†
‡28:3 MeV ÿ B A; Z†, as a function of the atomic number Z of the parent
nucleus. The experimental points are from cases where both parent nucleus and
daughter nuclei are þ-stable. The theoretical curve is from equation (4.5) (neglecting the pairing energies) together with equation (4.14).
compared with the total binding energies of the nuclei in this region. The
main deviation from the simple formula is due to the extra binding energy
of nuclei around the double-closed shell nucleus 208
82 Pb. This extra binding
energy not only makes these nuclei more nearly stable than average, but
also makes less stable the nuclei immediately above them. At higher Z,
around 238
92 U, there is another small region of relative stability.
From Fig. 6.1 it will be seen that þ-stable nuclei with Z > 66 (and a
few with Z 4 66) are in principle unstable to -decay. In practice the
decay rate is so low as to be almost unobservable if the energy release Q is
<4 MeV. Up to Bi Z ˆ 83† the lifetimes of þ-stable nuclei are many
orders of magnitude greater than the age of the Earth.
6.2
The theory of -decay
It is the electrostatic force which is responsible for inhibiting the -decay
of those nuclei for which the decay is energetically favourable. As an
example, consider the decay of bismuth to thallium,
76
Alpha decay and spontaneous fission
209
83 Bi
4
!205
81 Tl ‡ 2 He ‡ 3:11 MeV;
which is in principle possible but is not observed.
Figure 6.2 shows the electrostatic potential energy of the -particle
(charge 2e) at a distance r from a thallium nucleus (ZTl ˆ 81†,
2ZT1 e2 = 4"0 †r. Also indicated on the graph is the distance rs at which
the strong interaction with the thallium nucleus takes over, which we
estimate as
1
1
rs ˆ 1:1‰ 205†3 ‡ 43 Š fm ˆ 8:23 fm;
where we have used equation (4.2) for the radii of the Tl and He nuclei. It
is around this distance, in the surface region of the parent Bi nucleus, that
the -particle can be considered to be formed.
The graph immediately explains why the -particle ®nds it dif®cult to
escape even if it is formed. At rs the height of the Coulomb potential is
28.4 MeV, very much greater than the energy Q ˆ 3:11 MeV of the particle. Classically, it cannot penetrate the barrier and is free to move
only at distances greater than rc where rc is given by
Fig. 6.2 The potential energy of an -particle in the Coulomb ®eld of a thallium
nucleus, as a function of the separation distance r. At r rs the -particle from
the decay of bismuth is formed. At r ˆ rc it has penetrated through the classically
forbidden region.
6.2 The theory of -decay
Qˆ
2ZT1 e2
;
4"0 †rc
77
6:2†
rc is the classical distance of closest approach to the nucleus of an particle of energy Q coming from the outside. For thallium, rc ˆ 75 fm.
Thus classical mechanics forbids the -particle to escape. Quantum
mechanics, however, allows it to tunnel through, and we now estimate
this important tunnelling probability.
At distances r > rs , outside the range of the strong interaction, the
SchroÈdinger equation for the radial wave-function u r† of the -particle is
» 2 1 d2
ÿ
ru† ‡
2m r dr2
"
#
2Zd e2
»2 l l ‡ 1†
‡
u ˆ Qu:
4"0 †r 2m r2
6:3†
Here Zd is the atomic number of the daughter nucleus, and m is the
reduced mass, i.e.,
mˆ
m md
;
m ‡ md
6:4†
where m , md are the masses of the -particle and daughter nucleus. The
use of the reduced mass takes into account the recoil of the daughter
nucleus. To conserve angular momentum, the angular momentum of
the -particle and the angular momentum of the daughter nucleus must
combine to give the angular momentum of the parent. Also the parity of
the ®nal state, ÿ1†l (parity of daughter nucleus), must equal to the
ÿ
‡
parity of the parent nucleus. In the example Bi 92 † ! Tl 12 †, it is possible
to conserve angular momentum with l ˆ 4 or l ˆ 5 (Appendix C), but
parity conservation requires l ˆ 5.
For simplicity we shall only consider the case l ˆ 0. In fact the angular momentum term is usually small compared with the Coulomb potential. Writing u r† ˆ f r†=r, equation (6.4) then reduces to
ÿ
» 2 d2 f
2Zd e2
f ˆ Qf :
‡
2m dr2
4"0 †r
6:5†
If the Coulomb term were replaced by a constant potential V0 we should
have solutions
78
Alpha decay and spontaneous fission
f r† ˆ
8
>
< eikr ;
Q > V0 ;
>
: eKr ;
Q < V0 ;
2m
Q ÿ V0 †
»2
2m
K 2 ˆ 2 V0 ÿ Q†:
»
k2 ˆ
6:6†
This suggests that we try to ®nd solutions to equation (6.5) of the form
f r† ˆ e r† ;
6:7†
where r† is to be determined. By substitution, r† satis®es
!
"
2 # þ
»2 d2 d
2Zd e2
ÿQ :
‡
ˆ
2m dr2
4"0 †r
dr
6:8†
In a constant potential V0 , d=dr ˆ ik Q > V0 †, or K Q < V0 † and in
both cases d2 =dr2 ˆ 0. We shall assume that the Coulomb potential in
(6.8) is suf®ciently slowly varying for d2 =dr2 in this equation to be
neglected compared with d=dr†2 . Then
d
ˆ
dr
s
2m
»2
2Zd e2
ÿQ
4"0 †r
and
Z s
r† ˆ 2m
»2
2Zd e2
ÿ Q dr:
4"0 †r
6:9†
For r > rc , we can write our approximate solution in the form
Zr
Zr
k r†dr ‡ B exp ÿi
k r†dr ;
f r† ˆ A exp ‡i
rc
where
s
k r† ˆ ‡
2m
2Zd e2
;
Q
ÿ
4"0 †r
»2
and for rs < r < rc we can write
rc
6:10†
6.2 The theory of -decay
Z
f r† ˆ C exp ‡
rc
r
Z
K r†dr ‡ D exp ÿ
rc
r
79
K r†dr ;
6:11†
where
s
K r† ˆ ‡
2m
»2
2Zd e2
ÿQ :
4"0 †r
A, B, C, D are constants to be determined by the boundary conditions.
The solution (6.10) represents outgoing and incoming waves. The
incoming wave would be needed in an analysis of the scattering of particles by a nucleus. In the problem of -decay, only the outgoing wave
is present, so that B ˆ 0.
This solution must then be matched on to the exponentially increasing and decreasing functions included in (6.11). Any admixture of the
second term in the expression (6.11) quickly becomes negligible as r
decreases, since the fall of the exponential is very rapid for typical values
of the parameters, so that in the region rs < r < rc we may take
Z
f r† ˆ C exp
rc
r
K r†dr :
6:12†
At r ˆ rs , we must then match this solution to the appropriate radial
function of the -particle in the region where it is subject to the strong
interaction. Here, at the surface of the daughter nucleus, the description
of the -particle is really a very complicated many-body problem. But it is
reasonable to assume that in a heavy nucleus the rate of formation of particles is a property of the nuclear surface and does not vary greatly
from one nucleus to another. Given that an -particle has been formed,
the radial probability density of ®nding the particle at r ˆ rc relative
to the radial probability density of ®nding it at r ˆ rs is given by
4r2c ju rc †j2 j f rc †j2
ˆ
:
4r2s ju rs †j2 j f rs †j2
6:13†
We can therefore interpret
ÿ
ÿ
ÿf rc †ÿ2
ÿG
ÿ
ÿ
ÿf r †ÿ ˆ e ;
s
say;
6:14†
80
Alpha decay and spontaneous fission
as the transmission probability, through the Coulomb barrier, for -particles created at r rs . The essential correctness of this interpretation is
con®rmed by a more exact analysis of the wave-functions and matching
conditions.
Using equation (6.12)
Z
Gˆ2
rc
rs
K r†dr
s
2m 2Zd e2
ÿ Q dr
ˆ2
»2 4"0 †r
rs
r
Z
12
2mQ rc rc
ˆ2
ÿ 1 dr;
2
r
»
rs
Z
since rc ˆ
rc
2Zd e2
.
4"0 †Q
With the substitution r ˆ rc cos2 the integral is easily evaluated to give
r
G ˆ 2rc
r
2mQ
»2
Z
0
0
2 sin2 d
2mQ
ˆ 2rc
‰0 ÿ sin 0 cos 0 Š; where 0 ˆ cosÿ1
»2
þ
!s
2Zd e2
2mc2
ˆ
G rs =rc †;
Q
»c 4"0
"r #
rs
rc
6:15†
where the function
"
"r # r #
2
rs
rs
rs
ÿ1
cos
1ÿ
G rs =rc † ˆ
ÿ
rc
rc
rc
6:16†
is dimensionless and lies between 1 and 0 for 0 < rs =rc < 1 (Fig. 6.3). At
low energies rc ! 1 and G ! 1.
If the total ¯ux of -particles created at rs is 0ÿ1 , the probability per
unit time of -particle emission is 0ÿ1 eÿG , and hence the mean life for decay is given by
ˆ 0 eG :
6:17†
6.2 The theory of -decay
Fig. 6.3 The function G x† ˆ 2=†fcosÿ1
81
p
p
x ÿ ‰x 1 ÿ x†Šg.
We have argued that the rate of formation of -particles, and hence
0 , is a nuclear property unlikely to vary greatly from nucleus to nucleus.
In Table 6.1 we compare the formula (6.17) with experiment for a
sequence of naturally occurring -decays (all with l ˆ 0) initiated by
the most common isotope of uranium 238
92 U, taking 0 to be a constant.
ÿ23
The value 0 ˆ 7:0 10
s was chosen to give a reasonable ®t to this
sequence of measured lives. This value is not unreasonable on a nuclear
1
time scale of 2:6 10ÿ23 A3 s (}5.2), though it is somewhat shorter than
earlier estimates because we have chosen rs to be consistent with the
modern values for nuclear radii. Early workers obtained estimates of
nuclear radii from assumed values of 0 .
The qualitative agreement of the simple theory (which was proposed in 1928 by Gamow and by Condon and Gurney) with experiment is truly remarkable. The simple quantum-mechanical formula for
tunnelling comprehends time scales from as long as the age of the
Earth 1:45 1017 s† down to times less than a microsecond. The largest discrepancy between theory and experiment occurs with 210
84 Po; this
discrepancy can be associated with the closed shell N ˆ 126 in this
isotope.
82
Alpha decay and spontaneous fission
Table 6.1. The -decay series from
238
92 U
! 234
90 Th
238
92 U
rc
(fm)
Q
(MeV)
rs
(fm)
4.27
8.52
60.7
234
234
90 Th ! 91 Pa !
8.49
53.3
8.45
53.1
8.41
50.9
8.37
43.3
8.33
38.7
214
214
Pb
!
83 Bi !
82
8.28
30.1
210
210
82 Pb ! 83 Bi !
8.24
43.7
234
230
92 U ! 90 Th
230
226
90 Th ! 88 Ra
222
226
88 Ra ! 86 Rn
222
218
86 Rn ! 84 Po
214
218
84 Po ! 82 Pb
4.86
4.77
4.87
5.59
6.11
214
84 Po
! 210
82 Pb
7.84
210
84 Po
! 206
82 Pb
5.41
G
exp
(s)
0.53
2:0 1017 3:3 1017
0.51
0.51
0.50
0.46
0.43
214
84 Po†
0.36
210
84 Po†
0.47
1:1 1013
3:5 1012
7:3 1010
4:8 105
2:6 102
234
92 U†
theory
(s)
1:1 1013
3:9 1012
7:4 1010
4:2 105
1:6 102
2:3 10ÿ4 1:1 10ÿ4
1:7 107
5:8 105
The values of Q are from experiment. The intervening þ-decays, which reduce the
neutron-to-proton ratio as the nuclei become lighter, are given in parentheses.
This particular sequence of decays is of interest in the early history of
the study of radioactivity. As is indicated in Table 6.1, the daughter
nucleus of an -decay may be unstable to þ-decay. Since the mass number
A decreases by four in an -decay, and is unchanged in þ-decay, we expect
there to be three other similar sequences of -decays. Two of these, based
235
on 232
90 Th and 92 U, are also naturally occurring. The third, initiated by
237
93 Np, is made up of comparatively short-lived isotopes which must be
produced arti®cially for the series to be observed.
Since mean lives are dominated by the tunnelling factor, and this in
turn depends principally on the value of Q, we can now understand why
decay rates with Q < 4 MeV are so low.
It is found experimentally that -particle emission can take place with
the daughter nucleus left in an excited state. With even±even nuclei such
processes usually occur with much lower probability, since the value of Q
is reduced. However, the situation tends to be more complicated in the
case of even±odd and odd±odd nuclei. An unpaired nucleon is less likely
to take part in -particle formation, and its state may form part of an
excited con®guration of the daughter nucleus. In such a case, the daughter nucleus is likely to be found in this excited state after the emission of
an -particle.
6.3 Spontaneous fission
6.3
83
Spontaneous ®ssion
In -decay, a heavy nucleus splits into a light helium nucleus and another
heavy nucleus. Fission is the name given to a similar but more symmetric
process of a nucleus splitting into two more or less equal masses. The two
pieces are called ®ssion fragments.
The fragments are often nuclei in quite highly excited states, but we
can estimate the energy release in ®ssion by considering the simple case of
the symmetric ®ssion of an even±even nucleus A; Z† into two identical
nuclei A=2; Z=2† in their ground states, and using the semi-empirical
mass formula (4.5). We shall neglect the pairing energies. For a ®xed
ratio of protons to neutrons, the symmetry term as well as the cohesive
energy term is proportional to the total number of nucleons. Thus only
the surface and Coulomb energies contribute to the difference in binding
energy B of the two fragments and the parent nucleus:
B ˆ 2B A=2; Z=2† ÿ B A; Z†
2
2
ˆ ÿbA3 ‰2 12†3 ÿ 1Š ÿ
dZ 2
A
1
3
5
‰2 12†3 ÿ 1Š:
6:18†
If B is positive then this ®ssion is energetically possible, and the fragments will acquire kinetic energy B. From equation (6.18) nuclei for
which
2
Z 2 b 2 ÿ 23 †
;
>
2
A
d 23 ÿ 1†
i:e:
Z2
> 18;
A
6:19†
are metastable with respect to ®ssion. This condition is satis®ed by þstable nuclei heavier than 98
42 Mo. The energy release on the ®ssion of the
heavy elements is much larger than that in -decay. For example, the
energy release in the symmetric ®ssion of 238
92 U is ' 180 MeV. However,
although this is large, the process is strongly inhibited by the tunnelling
factor and spontaneous ®ssion is only observed in the heaviest of elements.
We envisaged the process of -decay as the initial formation of an particle at the surface of a nucleus and a subsequent tunnelling to freedom. It is not so easy to envisage the ®ssion process, or to calculate the
potential barrier. Figure 6.4 gives a schematic representation of the ®ssion
process, in which the nucleus is treated as a liquid drop. For the early
stages of the ®ssion of an initially spherical nucleus of radius R, it is
84
Alpha decay and spontaneous fission
Fig. 6.4 A schematic representation of a symmetric ®ssion in the liquid drop
model.
reasonable to consider the nucleus deforming into an ellipsoid of revolution as in the ®rst step of Fig. 6.4.
If we introduce a deformation parameter " such that
a ˆ 1 ‡ "†R
6:20†
1
b ˆ R= 1 ‡ "†2
where a, b are the major and minor semi-axes of the ellipsoid, the volume
of the drop 43†ab2 stays the same. It is not dif®cult to show that, for
small ", the surface area becomes to order "3
52 3
S "† ˆ 4R2 1 ‡ 25"2 ÿ 105
" †;
6:21†
and the surface energy will, correspondingly, increase. (For small ", the "3
term is a small correction.)
The Coulomb energy, on the other hand, decreases on deformation.
A (rather lengthy) calculation for a uniformly charged ellipsoid gives the
result
Ec ˆ
2 1
4"0 † 2
ZZ
d3 r d 3 r 0
jr ÿ r 0 j
3 Ze†2
4 3
1 ÿ 15"2 ‡ 21
" †
ˆ
5 4"0 R
6:22†
for small " (cf. equation (4.6)).
Using the parameters of the semi-empirical mass formula, these
results suggest that a small ellipsoidal deformation of a spherical nucleus
gives a change in energy of
þ
2
"
!
þ
!
2
2
2 23 1 Z 2
4
Z
3 52
bA ÿ d 1 ÿ "
bA3 ÿ d 1 :
5
5 A3
105
21 A3
6:23†
6.3 Spontaneous fission
85
The coef®cient of "2 is negative if
Z2 2b
>
ˆ 51:
A
d
6:24†
Hence when this condition is satis®ed the deformation energy is negative
even for small ", and ®ssion would proceed uninhibited by any potential
barrier. Thus (6.24) suggests there is an absolute upper limit for chemical
elements of Z ˆ 144 (using the relation between Z and A for þ-stable
nuclei given by equation (4.9)).
For elements of lower Z, spontaneous ®ssion involves tunnelling
through a potential barrier. We can crudely estimate the height of the
barrier from the expansion (6.23). For 235
92 U this gives the deformation
energy
83:35"2 ÿ 159:16"3 † MeV:
The coef®cient of "3 is negative, which con®rms that in the liquid drop
model the most likely deformation is indeed a prolate, rather than oblate,
ellipsoid, and the expression has a maximum of 3.4 MeV when " ˆ 0:35.
The measured potential barrier is 5.8 MeV. For A 240 barrier heights
are found to be between 5 and 6 MeV. Experimental values are determined from the threshold energies required to induce ®ssion, when the
nucleus is bombarded with, for example, ÿ-rays. Induced ®ssion by neutron capture also gives information on barrier heights. We shall consider
induced ®ssion in more detail in Chapter 8 and Chapter 9. It is a subject
of great technological importance.
As the inequalities (6.19) and (6.24) indicate, Z 2 =A is a measure of the
likelihood that a nucleus will be subject to spontaneous ®ssion. An
empirical, approximately linear, relationship exists between the logarithm
of the mean life for spontaneous ®ssion and Z2 =A. This is shown in Fig.
6.5 for some even±even nuclei.
The fragments produced in spontaneous ®ssion move apart rapidly
because of their Coulomb repulsion. They are neutron rich, since the
equilibrium neutron-to-proton ratio of a þ-stable nucleus decreases as
A decreases, and they are in highly excited states. Typically, one to
four neutrons `boil off ' from the fragments in a time of 10ÿ18 to
10ÿ15 s. Studies of the angular distribution of these `prompt neutrons'
show that they are indeed emitted from the moving fragments, rather
than at the moment of break-up of the ®ssioning nucleus. The resulting
86
Alpha decay and spontaneous fission
Fig. 6.5 Mean lives for spontaneous ®ssion of some even±even nuclei. (Data
from American Institute of Physics Handbook, 3rd ed., 1972, New York:
McGraw-Hill.)
nuclei are still far from the line of þ-stability. They reach their ground
states through the emission of prompt ÿ-rays and gradually decay, by þemission, to stable nuclei. It occasionally happens that a nucleus produced by þ-decay is unstable to neutron emission, and a `delayed neutron'
may result. For example, one of the ®ssion products of 236 U is 87
35 Br. This
þ-decays with a mean life of 80 s to either the ground state 87
36 Kr or an
87
excited state 36 Kr* which can lie above the threshold for neutron emis
86
sion. In the latter case the rapid decay 87
36 Kr ! 36 Kr ‡ n sometimes
occurs. Thus the time scale for the emission of the delayed neutrons is
determined by the long lifetimes of the þ-decay processes involved. We
shall see in Chapter 8 that such processes are important for the control of
nuclear reactors.
Problems
87
The semi-empirical mass formula predicts that the energy release in
spontaneous ®ssion is at a maximum when the two fragments are of equal
mass. Notwithstanding this, it is usually found that there is a quite striking asymmetry in the mass distribution of the ®ssion fragments. It is likely
that this asymmetry is due to shell structure effects. More detailed theories of ®ssion include these aspects of nuclear structure.
Problems
6.1
8
4 Be
decays to two -particles with a kinetic energy release of 0.094 MeV.
Estimate its mean life using the tunnelling formula (6.17), and compare
your estimate with the observed mean life of 2:6 10ÿ17 s.
6.2 The isotope
194
79 Au
undergoes þ-decay and has a mean life of 56 hours.
(a) One mode of decay is
194
79 Au
‡
! 194
78 Pt ‡ e ‡ ‡ 1:5 MeV:
The positron in this decay is created in the nucleus and must tunnel
through a Coulomb barrier to escape. Show that the barrier factor
suppresses the decay rate to a positron with an energy 1 MeV only
by a factor of about four.
(b) Another energetically possible decay, which has not been observed, is
194
79 Au
4
! 190
77 Ir ‡ 2 He ‡ 1:8 MeV:
Estimate the mean life for this mode of decay.
6.3
238
Pu decays by -emission:
238
94 Pu
! 234
92 U ‡
‡ 5:49 MeV,
with a mean life of 128 years. The mean life of 234 U is much longer, 2:5 105 years. Space probes to the outer planets use 238 Pu as a power source
for their equipment. Estimate the mass of 238 Pu needed to supply a
minimum of 1 kW of heat for 50 years.
6.4 The intermediate members of the radioactive series stemming from 238 U
have negligible mean lives on geological time scales (Table 6.1), so that
238
U may be said to decay to 206 Pb with a mean life of 6:48 109 years.
Similarly, 235 U decays to 207 Pb with a mean life of 1:03 109 years.
In a certain sample of uranium-bearing rock the proportions of atoms
of 238 U, 235 U, 206 Pb, 207 Pb were measured to be 1000:7.19:79.7:4.85. The
rock contained a negligible amount of 208 Pb, which is usually the most
common isotope of lead, indicating that all the lead in the rock came
from uranium decay. Estimate the age of all rock.
88
Alpha decay and spontaneous fission
6.5 Estimate the energy release and the velocity of the fragments in the
spontaneous ®ssion
238
92 U
119
! 119
46 Pd ‡ 46 Pd
where * denotes an excited state.
Neutrons `boil off' from the fragments. If in the frame of the moving
fragments the neutrons are emitted isotropically with energy 2 MeV,
describe qualitatively how the neutrons appear in the laboratory.
7
Excited states of nuclei
7.1
The experimental determination of excited states
So far we have for the most part been considering atomic nuclei in their
quantum ground states. Most nuclei on Earth have been in their ground
states since the time of its creation. However, almost all nuclei possess
excited states of higher energy (and therefore less binding energy) than
their ground state. There are many ways of exhibiting these excited states,
and determining their energies and quantum numbers. One method is to
scatter energetic protons of known momentum pi from the nucleus of
interest and to observe their angle of scattering and ®nal momentum
pf . This process is illustrated in Fig. 7.1.
To conserve momentum, the recoiling nucleus has momentum
pi ÿ pf cos † in the direction of the incoming proton and pf sin in the
perpendicular direction. Taking all momenta and energies to be nonrelativistic, the difference E between the initial and ®nal kinetic energies
of the system is
Eˆ
p2i
p2
p2 ‡ p2f ÿ 2pi pf cos †
ÿ f ÿ i
;
2mp 2mp
2mA
7:1†
where mA is the mass of the recoiling target nucleus. By conservation of
energy E must be the excitation energy given to the nucleus. In terms of
the initial and ®nal proton kinetic energies Ei and Ef , equation (7.1)
becomes
89
90
Excited states of nuclei
Fig. 7.1 Scattering of a proton from a nucleus initially at rest.
mp
mp
2mp
1
E ˆ Ei 1 ÿ ÿ Ef 1 ‡ ‡ Ei Ef †2 cos :
mA
mA
mA
7:2†
In equations (7.1) and (7.2), mA ˆ mA ‡ E=c2 may be replaced by the
mass mA of the nucleus in its ground state, with little error.
In practice, a mono-energetic beam of protons is directed at a target
containing the nucleus in question. If the target is a solid it is generally
made so thin that the probability of a proton scattering more than once
off a nucleus is small.
At a ®xed scattering angle the emerging protons are no longer
mono-energetic but, apart from a background coming from, for example,
the residual multiple scattering, their energies fall into several well-de®ned
peaks. An example of this is shown in Fig. 7.2.
Fig. 7.2 The number of protons scattered at 908 from a static target containing
10
B, as a function of their ®nal energy Ef . Initially the protons were in a collimated beam and had energy 10.02 MeV. Background scattering has been
removed. (Data from Armitage, B. H. & Meads, R. E. (1962), Nucl. Phys. 33,
494.)
7.1 The experimental determination of excited states
91
In the experiment from which this data is taken, initial protons of
energy 10.02 MeV were scattered from 105 B, and the graph shows the
number of protons scattered within a small angular range at ˆ 908 as
a function of their ®nal energy Ef . The peak of the highest energy at
Ef ˆ 8:19 MeV corresponds to elastic scattering, since equation (7.2)
then gives E ˆ 0, that is, no excitation. The values of Ef for the successive
peaks of lower energy give a sequence of excitation energies E of the 105 B
nucleus (Problem 7.1).
The area under the peak at a particular Ef in Fig. 7.2 is proportional
to the probability of producing the corresponding excited state. This
probability depends both on Ei and on . Information on the spin and
parity of the state can be obtained from measurements of the angular
dependence of the production probability. Further information on spin
and parity is given by the energies and angular distribution of ÿ-rays that
can result as the excited states decay back to the ground state.
The inelastic scattering of protons as in the above example is a technique which may be used with nuclei which are not radioactive and which
can be safely made into targets. Another technique, which is suitable for
determining the energy levels of some þ-unstable nuclei also, is that of
deuteron stripping.
In deuteron stripping, a mono-energetic beam of deuterons is directed
at a target nucleus. As well as elastic and inelastic deuteron scattering,
leaving the original, possibly excited, target nucleus, a nuclear reaction
may take place in which the deuteron loses a nucleon into the target
nucleus. Consider for example the reaction represented by
2
1H
A‡1 ‡A
Z X ! Z X ‡ p;
7:3†
in which only the proton emerges.
A‡1
Here A
Z X is the target nucleus and Z X is its isotope (perhaps
unstable) with one more neutron. The * denotes a possible excited state.
If the emerging proton in this reaction is at an angle with respect to
the beam of incident deuterons and has energy Ef , a calculation similar to
that for proton scattering yields for the excitation energy of the ®nal
A‡1 X , the expression
nucleus, Z
1
mp
mp md Ei Ef †2
m
cos ‡ E0 ;
E ˆ Ei 1 ÿ d ÿ Ef 1 ‡ ‡2
mA‡1
mA‡1
mA‡1
7:4†
92
Excited states of nuclei
Table 7.1
Ef (MeV)
E (MeV)
11.42
5.08
11.97
4.56
12.69
3.85
13.50
3.06
15.74
0.87
16.62
0.0
This shows the mean energies Ef of groups of protons that emerge, from a static
target containing 16 O, at an angle of 198 to a 14.95 MeV deuteron beam. Below
are the corresponding excitation energies E of 17 O, as calculated using equation
(7.4). (Data from Yakgi, K. et al. (1963), Nucl. Phys. 41, 584.)
where now Ei is the incident deuteron energy, md is the deuteron mass
and E0 ˆ mA ‡ md ÿ mA‡1 ÿ mp †c2 is the difference in rest mass energies
between the initial and ®nal nuclei in their ground states.
Table 7.1 shows the results of a deuteron-stripping experiment,
2
1H
‡ 168 O ! 178 O ‡ p;
in which a deuteron beam with energy Ei ˆ 14:95 MeV was directed at a
target containing 168 O and the energies of protons detected at ˆ 198 were
measured. In this example E0 ˆ 1:93 MeV. The table shows the six proton groups with the highest energies and the corresponding 178 O excitation
energies E. The highest-energy proton group with Ef ˆ 16:62 MeV corresponds to the production of 17 O in its ground state.
Figure 7.3 shows the excitation energies of 17 O up to 6 MeV. In this
energy-level diagram the excited states are denoted by horizontal lines at a
height above the ground state that is proportional to the excitation
energy. The ®ve lowest excited states are those determined from the
above deuteron-stripping reaction. The experimental information on
the others will be discussed in }8.1. We shall in general restrict our discussion to energy levels below about 10 MeV, which is the most important energy range for the topics we discuss in later chapters.
Also shown on the energy-level diagram are the lowest energies,
called threshold energies, such that excited states above these thresholds
can break up into the smaller nuclei indicated. These energies are computed from the masses of the nuclei involved. The lowest threshold is for
17
O to disintegrate into 16 O and a neutron. Below this threshold the
excited states cannot disintegrate into lighter nuclei but they decay electromagnetically, for example by the emission of a photon, to lower energy
states and, eventually, the ground state.
7.2 Some general features of excited states
93
Fig. 7.3 The 17 O energy-level diagram up to an excitation energy of 5.94 MeV.
The ®rst ®ve excited state energies are as determined from deuteron stripping
(Table 7.1). Also shown is the threshold energy at 4.15 MeV for break-up into
a neutron and 16 O (the `neutron separation energy' of 17 O), and the threshold
energy for break-up into 13 C and an -particle. (For more information see
Ajzenberg-Selove, F. (1982), Nucl. Phys. A375, 1.)
The spins and parities of the excited state, some of which are shown
on the diagram, are deduced from measurements of the angular distribution of the protons from the nuclear reactions which produce the states,
and also from the angular distributions of the photons resulting from the
subsequent decays of the states.
7.2
Some general features of excited states
In general, the heavier the nucleus the more excited states it has. The
deuteron has no excited states and very light nuclei have only a few
well-de®ned excited states. However, the number of excited states
increases rapidly as A increases. Figure 7.4 gives the energy levels up to
9 MeV of the two light nuclei 115 B and 116 C. This pair is an example of socalled mirror nuclei: the number of protons in either one equals the num-
94
Excited states of nuclei
Fig. 7.4 Energy-level diagrams for the mirror nuclei 115 B and 116 C. The spins and
parities of the states are also given. Note the proton separation energy from 116 C at
8.69 MeV, and the -particle separation energies. (Data from Ajzenberg-Selove,
F. & Busch, C. L. (1980), Nucl. Phys. A336, 1.)
ber of neutrons in the other. The near equality of their energy levels
illustrates the charge independence of the strong force; for this pair of
light nuclei the difference in Coulomb energies is small and the nuclear
physics is almost identical.
A qualitative understanding of the excited states is given by the shell
model. Consider the 115 B nucleus. The six neutrons ®ll the 1s12 and 1p32
shells. There are two protons ®lling the 1s12 shell and in the 1p32 shell two
protons have their angular momenta coupled to zero while the odd
ÿ
remaining proton gives the ground-state spin and parity 32 . The ®rst
7.2 Some general features of excited states
95
ÿ
excited state, spin and parity 12 , can be considered within the shell model
to be the state in which the odd proton is taken from the 1p32 shell and
placed in the higher energy 1p12 shell. Such a state is known as a singlenucleon excitation.
Many of the higher energy states will correspond to several nucleon
excitations. The fact that there is a large number of excited states is easily
accommodated within the shell model. If we consider only the 1p32 and 1p12
shells, the four neutrons can be distributed in ( 64 ) ways over the six
available single-particle neutron states, and the three protons in ( 63 )
ways over the single-particle proton states. Thus we can construct
( 64 ) ( 63 ) ˆ 15 20 ˆ 300 independent states ± more than enough to
account for all of the states of negative parity below the -decay threshold, even allowing for the fact that levels with spin j have 2j ‡ 1†
members.
Figure 7.5 shows energy-level diagrams for two heavier nuclei, 46 Ca
and 108 Pd. Note in these examples that for a given excitation energy, the
heavier nucleus has a greater density of excited states and, for a given
nucleus, the density of states increases as the excitation energy increases.
These qualitative features are apparent in most nuclei, though near to
Fig. 7.5 Energy-level diagrams for 46 Ca and 108 Pd. (Data from Nuclear Data
Sheets of the National Nuclear Data Centre for Nuclear Data Evaluation, Sheet
37 (1982), 290; Sheet 38 (1983), 467: Academic Press.)
96
Excited states of nuclei
closed-shell nuclei the energy gaps between levels tend to be signi®cantly
greater, especially at low excitation energies. Again, the shell model provides an explanation. The elementary formula (5.4) for the integrated
3
density of single-nucleon states gives N E† E 2 for neutrons or protons,
so that the number of single-nucleon states N in a small energy range
E is given by
N dN 3 N
ˆ
:
E
dE
2E
7:5†
Hence, taking N ˆ 1, the mean spacing between single-particle neutron
levels at the Fermi energy EF 38 MeV; N EF † ˆ N; see }5.2) is
E ˆ
2 EF 25
MeV;
3N
N
7:6†
with a similar result for the protons. E very largely sets the energy scale
for the excited states, so that as N A=2† increases they come closer
together.
In the shell model, the lowest-lying excitations can often be associated
with single-particle excitations. At higher energies, several nucleons can
be simultaneously excited, and the increasing density of states with energy
re¯ects the increasing number of possible con®gurations involving many
excited nucleons.
Often, such complex nuclear states can be quite simply described by
models which naturally incorporate multi-particle motion. For example,
the liquid drop with which we started our discussion of nuclei, and
which we deformed in our discussion of spontaneous ®ssion, can be
envisaged to be in an excited state of vibration or one of overall rotation. Although we will not dwell here on these interesting and useful
models, many excited states which it would be clumsy to describe in
terms of the shell model can be justi®ably envisaged as vibrational and/
or rotational states.
The excited states of nuclei are not stable. Their energies, being of the
order of MeV for light nuclei and keV for heavy nuclei, are so high they
play an insigni®cant role in terrestrial thermodynamics. At temperatures
accessible in laboratories they decay to states of lower energy and ultimately to the ground state. We now take up the question of their modes
of instability and their mean lives.
7.3 The decay of excited states: ÿ-decay and internal conversion
7.3
97
The decay of excited states: ÿ-decay and internal conversion
Excited states that have energies below the lowest threshold for break-up
into lighter nuclei decay almost exclusively electromagnetically. The most
prominent mode is ÿ-decay, in which the nucleus changes to one of its
lower energy states and simultaneously emits a single photon. A nucleus
can also decay by internal conversion, which is a process whereby electromagnetic energy liberated by the nucleus is taken up by an atomic electron which is ejected. The energy of the emitted particle, be it photon or
electron, is the energy lost by the nucleus, with corrections for small recoil
effects and, in the case of internal conversion, the electron's atomic binding energy.
Electromagnetic mean lives can be as long as hundreds of years, or
as short as 10ÿ16 s. The transitions are slow if the change in nuclear spin
is large. To understand this great disparity in decay rates it must be
appreciated that photons, like other particles, have angular momentum,
which is the sum of their intrinsic and orbital angular momentum. The
intrinsic photon spin is one, so that the total angular momentum quantum number j of a photon is integral. The allowed values are
j ˆ 1; 2; 3; . . .; the value j ˆ 0 is not possible: photons do not exist in
states of zero total angular momentum (just as classically, since electromagnetic waves are transverse, it is impossible to construct wave-like
solutions of Maxwell's equations with spherical symmetry). If the
nucleus changes its spin from ji to jf in a ÿ-decay, then to conserve
angular momentum
ji ‡ jf 5j 5j ji ÿ jf j;
as is shown in Appendix C. Thus ÿ-ray transitions between states with
ji ˆ 0 and jf ˆ 0 are absolutely forbidden (but transitions by internal
conversion are possible). It may be shown theoretically that transition
rates are much suppressed as j increases; the theory of ÿ-decay and internal conversion will be discussed more fully in Chapter 12.
As well as angular momentum, parity is conserved in electromagnetic
transitions. The photon parity must be positive if the initial and ®nal
states have the same parity and negative if they have opposite parities.
A photon has parity ÿ1† j when the decay is `electric' with the nucleus
basically coupling to the electric ®eld of the photon, and parity ÿ ÿ1† j
when the decay is `magnetic' with the nucleus coupling to the magnetic
®eld of the photon.
98
Excited states of nuclei
Figure 7.6 shows the results of rough theoretical estimates of ÿ-decay
rates. Precise calculations require a detailed knowledge of the initial and
®nal nuclear wave-functions, which is not generally available. As an
example, consider the decay of the ®rst excited state of 178 O (Fig. 7.3).
This can only decay to the ground state and, neglecting internal conversion, will do so by emitting an 0.87 MeV photon. (See Problem 7.3 for
recoil effects.) The nuclear spin changes from 12 to 52 and there is no change
in nuclear parity. Therefore the photon must have positive parity and
j 5j 12 ÿ 52 j ˆ 2. The value j ˆ 2 is the most likely photon angular momentum; the value j ˆ 3 is possible but would give a much lower decay rate.
The experimentally observed mean life is 2:58 0:04† 10ÿ10 s, in fair
agreement with the value suggested by Fig. 7.6 for an electric transition
with j ˆ 2.
Fig. 7.6 Estimated mean lives for electric multi-pole radiation of order 2 j as a
function of the energy of the emitted photon, for a nucleus with A ˆ 100.
Corresponding estimates for other nuclei may be obtained by multiplying by
100=A†2j=3 . Mean lives for magnetic multi-pole radiation are generally longer
than those for electric multi-pole radiation of the same order by a factor
2
M =E 20A3 .
(The lines are drawn from formulae given, for example, in Jackson, J. D. (1975),
Classical Electrodynamics, 2nd ed., New York: Wiley, p. 760.)
7.4 Partial decay rates and partial widths
99
Measurements of photon energies clearly give information on the
energies of excited states, and such measurements have played a large
part in determining these energies. Measurements of decay rates and of
the angular distributions of the intensity and polarisation of the photons
give information on the `multi-pole' type of the transition. Transitions
with j ˆ 1; 2; 3; . . . ; n; . . ., are referred to as dipole, quadrupole, octapole,
. . . ; 2n -pole, . . ., transitions; each type of transition has its characteristic
lifetime and angular distribution. Unravelling the multi-pole type of a
transition is one of the ways of determining the spins and parities of
the nuclear states involved. Long-lived excited states of nuclei are
known as isomeric states.
7.4
Partial decay rates and partial widths
In general, an excited state of a nucleus has the option of decaying in
several ways. There may be several lower energy states to which it can
decay by ÿ-emission, or it may be able to break up into lighter nuclei. For
example, the 4.56 MeV excited state of 17 O (Fig. 7.3) can decay by neutron emission, or by ÿ-emission to any one of four lower energy levels.
With each mode of decay, or decay channel, say the ith, there will be a
partial decay rate 1=i , and the total decay rate 1= is simply the sum of
the partial decay rates:
1 X1
:
ˆ
i
i
7:7†
is the mean lifetime of the excited state (}2:3†:
The partial width of the ith channel is de®ned to be ÿi ˆ »=i and the
total width ÿ ˆ »=, so that
ÿˆ
X
i
ÿi :
7:8†
The ÿi and ÿ have the dimensions of energy. It is shown in Appendix D
that an excited state does not have a de®nite energy, but a distribution of
energies of width ÿ about a mean energy E. Hence the relation
ÿ ˆ »
7:9†
100
Excited states of nuclei
can be interpreted as a relation between the uncertainty in energy of a
state and its lifetime, rather like the Heisenberg uncertainty relation
between position and momentum of a particle.
The particle decay rates of nuclei for ÿ-emission are rarely greater
than 1016 sÿ1 . The corresponding partial widths are thus generally less
than about 5 eV (and the energies of excited states that decay only by ÿemission, expressed in MeV, can be quoted to ®ve decimal places).
7.5
Excited states arising from þ-decay
When a þ-unstable nucleus decays, it may be energetically possible for the
transition to be to an excited state of the daughter nucleus. Although the
immediate energy release for decay to an excited state is less than that for
decay to the ground state, there are many þ-decays in which the selection
rules discussed in }12.2 and }12.6 make decay to an excited state more
likely. The excited state will then itself decay, usually by ÿ-emission.
As an example, Fig. 7.7 shows the decay scheme of 60
27 Co, which is þunstable with a mean life of 7.6 years. 60
27 Co rarely decays directly to the
ground state of 60
28 Ni, but with 99.9% probability it decays to a state with
an excitation energy of 2.50 MeV. The þ-emission is quickly followed by
the emission of two photons with energies of 1.17 MeV and 1.33 MeV,
Fig. 7.7 The þÿ decay of 60 Co illustrated with energy-level diagrams. The decay
takes place predominantly to a state of 60 Ni with excitation energy 2.50 MeV,
sometimes to a state with excitation energy 1.33 MeV, and rarely to the ground
state. The spins and parities of the states are also given.
Problems
101
giving a total ÿ energy of 2.50 MeV. In almost all of the remaining 0.1%
of þ-decays, the electron emission is followed by a single-photon emission
of energy 1.33 MeV. Thus there must be two excited states of 60
28 Ni
involved in these processes, ordered as shown in the ®gure.
60
Co has important uses in medicine and technology as a source of ÿrays. It is manufactured by the irradiation of natural 59 Co in a nuclear
reactor.
An extensive and detailed compilation of data on nuclear energy
levels will be found in Firestone, R. B., Shirley, V. S., editor (1996),
Table of Isotopes, 8th ed., New York: John Wiley.
Problems
7.1 From the data given in Fig. 7.2 draw an energy-level diagram for the
nucleus 105 B.
7.2 Derive equation (7.4).
7.3(a) Using the data of Table 7.1, show that the recoil velocity of a
nucleus produced in its ®rst excited level is
v ˆ 5:7 10ÿ3 c
17
O
E0 ˆ 1:918 MeV†:
(b) If this 17 O nucleus comes to rest before it decays, show that the energy
of the emitted photon is about 24 eV less than the excitation energy of
the nucleus.
(c) If the photon is emitted from the moving nucleus, show that because of
the Doppler effect it will be changed in energy by between ÿ5 keV and
5 keV.
7.4 The binding energies of the mirror nuclei 115 B and 116 C are 76.205 MeV
and 73.443 MeV respectively. Assuming that the difference is due
entirely to Coulomb effects, and that the proton charge is uniformly
distributed through a sphere of radius RC in both nuclei, ®nd RC .
This was an early way of estimating the size of a nucleus. Compare
1
RC with the value R ˆ 1:12A3 fm, and comment on the difference.
7.5 The excited state 17 O at 4.56 MeV (Fig. 7.3) has a mean life of only
1:6 10ÿ20 s. How can this be so short? Estimate the width ÿ of the
state.
7.6 What type of electromagnetic transition do you expect between a state at
2.13 MeV in 115 B (Fig. 7.4) and the ground state? Estimate the mean life
of this state.
7.7 Consider the energy levels of 105 B (Problem 7.1). The ground state has
spin and parity 3‡ , and the excited states in order of increasing excita-
102
Excited states of nuclei
tion energy are 1‡ ; 0‡ ; 1‡ ; 2‡ ; 3‡ ; 2ÿ ; 2‡ ; . . . . Is there an explanation
within the shell model of why the lowest states all have positive parity?
The ®rst excited energy level is at 0.72 MeV and the second at
1.74 MeV. Given a large number of nuclei in the second excited state,
what energies have the ÿ-rays that result from the decays? Estimate the
relative numbers of these ÿ-rays.
8
Nuclear reactions
In a nuclear reaction two nuclei, or a nucleon and a nucleus, come
together in such close contact that they interact through the strong
force. The deuteron-stripping reaction of equation (7.3) is one example.
A reaction which contributes to energy generation in stars is
16
O ‡ 16 O ! 28 Si ‡
‡ 9:6 MeV;
8:1†
and a nuclear reaction important in power technology is
n ‡ 235 U ! fission products:
The latter two are both exothermic reactions in which the kinetic energy
of the ®nal nuclei is greater than that of the initial nuclei. In an endothermic reaction energy must be supplied before the reaction will take place,
as in the reaction inverse to (8.1) above.
8.1
The Breit±Wigner formula
The concept of cross-section (Appendix A) is important for understanding
and classifying nuclear reactions. Figure 8.1 shows the total cross-section
for neutrons to interact with the 168 O nucleus as a function of the kinetic
energy E (in the centre-of-mass system) up to E ˆ 2:3 MeV. The principal features of the cross-section are the high but narrow resonance peaks,
superposed on a slowly varying background. These peaks are due to the
formation of excited states of 17 O from the neutron and 16 O at the reso103
104
Nuclear reactions
Fig. 8.1 The total cross-section for neutrons interacting with 16 O as a function of
centre-of-mass energy, showing resonances that correspond to the formation of
excited states of 17 O (top scale: see also Fig. 7.3). (Data from Garber, D. I. &
Kinsey, R. R. (1976), Neutron Cross Sections, vol. II, Upton, New York:
Brookhaven National Laboratory.)
nance energies. When the energy of the incident neutron is such that the
total energy of the system matches, to within the width ÿ, one of the
excited states energies of 17 O, the neutron is readily accepted into the
target to form that state. Note that the binding energy of the neutron
in the ground state of the so-called compound nucleus becomes available
as excitation energy. In our example of 17 O, if excitation energies are
measured from the ground state the neutron binding energy of
4.15 MeV (cf. Table 4.2 and Fig. 7.3) has to be added to the resonance
energies to obtain the 17 O excitation energies. This displaced energy scale
is also given in Fig. 8.1. Thus only those excited states above 4.15 MeV
can appear in the data.
The six peaks which appear in Fig. 8.1 correspond to the top six levels
of the energy-level diagram, Fig. 7.3. The two lowest of these six correspond to states found in the deuteron-stripping reaction we discussed
earlier in }7.1.
It is shown in Appendix D that excited states make a contribution to
the total cross-section in the neighbourhood of the resonance energy E0
of approximately the form
8.1 The Breit±Wigner formula
tot E† ˆ
gÿi ÿ
;
k2 E ÿ E0 †2 ‡ ÿ2 =4
105
8:2†
where k ˆ jkj, and k is the wave-vector of the incoming neutron in the
centre-of-mass frame, ÿi is the partial width for decay into the incident
channel 16 O ‡ n, g is a statistical factor (in this case g ˆ 2j ‡ 1†=2 where j
is the spin of the excited state). The expression (8.2) is known as the Breit±
Wigner formula. For ÿ 5 E0 the cross-section is at a maximum when
E ˆ E0 , and falls to half its maximum value at E ˆ E0 ÿ=2. Thus ÿ
is the `full width at half-maximum' of the peak. The peak width seen
experimentally depends also on the energy spread of the incident neutron
beam (no particle beam is ever perfectly mono-energetic), on the thermal
motion of the nuclei in the target, and on the characteristics of the detectors, so that a careful analysis may be necessary before a true intrinsic
width can be obtained from the raw experimental data.
Consider the peak at E ˆ 0:41 MeV in Fig. 8.1. The estimated width
of the peak, ÿ 0:04 MeV, corresponds to a mean life of 1:6 10ÿ20 s.
This is short compared with mean lives for ÿ-emission, but still quite long
on the nuclear time scale (}5.2) of the oxygen nucleus of 10ÿ22 s. Such a
long mean life can be understood as resulting from the nature of the
excited state, which is a compound nuclear state in which many nucleons
participate. The neutron entering the nucleus loses its energy by collisions
with other nucleons, and if it loses more than 0.41 MeV it can no longer
escape. The nucleus then stays in the excited state until such time as a
single neutron again acquires enough energy to get away, or (in this case
with much lower probability) the nucleus decays electromagnetically. In
the latter case, if the decay is to the ground state or any other state below
the neutron separation energy, the neutron is captured; this process is
known as radiative capture.
In between resonance peaks, an analysis of the background crosssection suggests that the nucleus resists penetration by the incident neutron. The neutron appears to be repelled from the surface of the 16 O
nucleus at energies off resonance.
Resonance peaks are a feature of all cross-sections for neutron scattering from nuclei with A 5 4 and neutron energies up to a few MeV. The
`binding energy of the last neutron' (or separation energy: see equation
(5.6)) which is available for excitation energy usually lies in the range
5 MeV±15 MeV. As explained in }7.2, the density of excited states at
®xed energy increases rapidly with A. Thus for neutron scattering from
heavy nuclei the number of resonances per MeV increases rapidly with A.
106
Nuclear reactions
Also, as A increases, the width of the states becomes narrower: the states
become more stable since in the compound nucleus the incoming neutron
has more nucleons with which to share its energy, and the probability of
any one of them acquiring enough energy to escape decreases.
All this is illustrated in Fig. 8.2, which shows the total cross-section at
low energies for neutrons interacting with the heavy nucleus 238
92 U. Note
that the horizontal energy scale is in electron volts, and the vertical crosssection scale is logarithmic. The resonance peaks are associated with the
formation of excited states of 239 U, and the spacings between the peaks
are only 20 eV. The resonances are very narrow, with an intrinsic width
of order 10ÿ2 eV. Indeed, the states are here so narrow that ÿ-decay
competes signi®cantly with other decay modes, and roughly half of the
decays of the excited states formed at these resonances are electromagnetic and result in radiative capture. The other prominent decay mode is
neutron emission. Less-common modes include -decay and ®ssion.
For neutron energies that are off resonance the cross-section of Fig.
8.2 is dominated by the neutron scattering from the surface of the 238 U
nucleus. However, other nuclear reactions are energetically possible and
may occur. For example, the neutron could pick up two protons and
another neutron from the 238 U surface to form an -particle:
4
235
n ‡ 238
92 U ! 2 He ‡ 90 Th:
Fig. 8.2 The total cross-section for neutrons interacting with 238 U, as a function
of centre-of-mass energy. Note that the vertical scale for the cross-section is
logarithmic and the horizontal energy scale is in electron volts. (Data as in Fig.
8.1.)
8.2 Neutron reactions at low energies
107
Such a reaction, when the neutron energy is off resonance, does not
proceed through the formation of 239
92 U, and is known as a direct nuclear
reaction.
8.2
Neutron reactions at low energies
Since neutrons are uncharged there is no Coulomb barrier to overcome;
hence neutrons of very low energy easily penetrate matter and interact
with nuclei. In the limit E ! 0, only elastic scattering and exothermic
nuclear reactions can take place. When a nuclear reaction is possible it
can be expected that the reaction rate at suf®ciently low energies will be
independent of E, and simply proportional to the density of neutrons in
the neighbourhood of the nucleus. The cross-section ex for exothermic
nuclear reactions is given (Appendix A) by
(neutron flux) ex ˆ reaction rate per nucleus.
The neutron ¯ux is n v (where n is the neutron number density in the
beam and v is the velocity of the neutrons relative to the target nucleus).
Since the right-hand side of the equation is also proportional to n , it
follows that
ex ˆ
constant†
v
8:3†
at suf®ciently low energies. This is the behaviour that is seen experimentally.
If the low-energy region lies in the wing of a resonance, the 1=v† law
follows from the Breit±Wigner formula (8.2). In this case, we must take
into account the energy dependence of the partial width ÿi E†, found in
equation (D.9) of Appendix D. ÿi E† contains the factor ni E†, which is
proportional to k (from equations (B.6) and (B.8) of Appendix B). For
E 0, the Breit±Wigner formula then gives
1 constant†
:
k E02 ‡ ÿ2 =4
8:4†
Since »k ˆ mv, where m is the reduced mass of the neutron and target
nucleus, we recover the 1=v† law. The combination of the 1=v†, or,
108
Nuclear reactions
1
equivalently, 1=E 2 † law with a low-lying resonance is well illustrated in
Fig. 8.3, which shows the low-energy cross-section for cadmium.
In quantum mechanics, non-elastic processes are always accompanied by elastic scattering, just as, in optics, absorption is always accompanied by diffraction. The elastic scattering of neutrons by nuclei takes
place through compound nucleus formation and by surface scattering;
the two processes are not independent and must be considered together.
It may be shown that the elastic-scattering cross-section of slow neutrons does not follow the 1=v† law but tends to a constant value as
E ! 0. This limiting value depends sensitively on the presence of resonances near threshold. If the target nucleus has spin, the cross-section
also depends on the relative orientation of the spins of the neutron and
nucleus.
Fig. 8.3 The total cross-section for neutrons interacting with natural cadmium.
The open circles are experimental points (data as in Fig. 8.1). The line is a ®t
with
tot ˆ
constant†
;
v‰ E ÿ E0 †2 ‡ ÿ2 =4Š
taking E0 ˆ 0:18 eV and ÿ ˆ 0:12 eV. Note that both the scales are logarithmic.
On a logarithmic plot the `1=v' form at very low energies gives a straight line with
1
a slope of ÿ 12 (cf. equation (8.4), v ˆ 2mE†2 †. This is evident for E < 0:03 eV.
The very large resonance cross-section is due to 113 Cd, which constitutes 12.3% of
natural cadmium.
8.3 Coulomb effects in nuclear reactions
8.3
109
Coulomb effects in nuclear reactions
Our discussion of nuclear reactions has so far emphasised reactions involving neutrons. In a nuclear reaction involving a proton and a nucleus, or
two nuclei, there are seen the same features of resonance scattering with
the formation of a compound nuclear state, and direct nuclear reactions
off resonance.
However, the effect of the Coulomb repulsion between particles in the
initial or ®nal channels of the reaction leads to signi®cant differences in
the reaction cross-sections at low energies below the Coulomb barrier
height. This effect is illustrated in Fig. 8.4, which shows the low-energy
cross-section for the nuclear reaction
‡ 136 C ! n ‡ 168 O;
Fig. 8.4 The cross-section for the reaction ‡ 13 C ! n ‡ 16 O. The dashed curve
ÿ ÿ ÿ exhibits the large Coulomb suppression at low energies (see text). Note the
resonances at high excited-state energies of 17 O (top scale) which are above those
shown in Fig. 7.3. (Data from Blair, J. K. & Haas, F. X. (1973), Phys. Rev. C7,
1356.)
110
Nuclear reactions
as a function of the centre-of-mass kinetic energy E of the incident nuclei.
The reaction is in fact exothermic, with an energy release of 2.2 MeV
(Fig. 7.3) and so it can in principle occur at any energy. However, at
low energies the -particle must tunnel through a Coulomb barrier before
it can interact with the 13 C nucleus. The barrier is about 4 MeV high. In
classical mechanics a nuclear interaction could not occur for an -particle
having lower energy than this. In quantum mechanics the particle can
tunnel through, but the low-energy cross-section is much suppressed, as
the ®gure clearly demonstrates. The tunnelling probability for the -particle to penetrate the barrier from the outside is the same as the probability for tunnelling in the other direction, as in -decay, and this we
estimated in Chapter 6 to be eÿG E† , where G E† is given by equation
(6.15) (but with Q replaced by E).
It is usual to parametrise charged-particle reaction cross-sections at
low energies by the expression
E† ˆ
1
S E†eÿG E† ;
E
8:5†
and Fig. 8.4 also shows this curve with S E† chosen to be a constant 0.3
barn MeV to ®t the cross-section at the lowest energies. The background
cross-section below the resonances roughly follows this curve, but large
resonance peaks due to the formation of excited states of 17 O are evident.
The precise form of charged-particle nuclear reaction cross-sections
at low energies is of great importance, both in astrophysics and for the
prospect of controlled thermonuclear reactions on Earth. It may be
shown that, as E ! 0, the function S E† in (8.5) tends to a constant
value, which depends on the particular reaction and is very sensitive to
the proximity of resonances. We can give a qualitative derivation of this
result for the case when the low-energy region lies in the wing of a nearby
resonance. The Breit±Wigner formula (8.2) may be written
E† ˆ
»3
gÿ ÿi = »†
;
2mE E ÿ E0 †2 ‡ ÿ2 =4
8:6†
where m is the reduced mass of the interacting particles and ÿi = » is the
decay rate into the incident channel. For energies close to threshold it is
again important to include the energy dependence of ÿi . Recalling the
discussion of -decay in Chapter 6, we replace the decay rate ÿi = »† by
8.4 Doppler broadening of resonance peaks
111
1=0 †eÿG E† , where 0 is a constant nuclear time. For E 0 the expression
(8.6) then reduces to
E† ˆ
»3
gÿ
eÿG E†
:
2m0 E02 ‡ ÿ2 =4 E
This is of the same form as (8.5) with
S E† ˆ
»3
gÿ
;
2
2m 0 E0 ‡ ÿ2 =4
a constant.
If the nuclei in a reaction have charges Z1 e, Z2 e, the expression (6.15)
for G E† must of course be generalised slightly: 2Zd e2 is replaced by
Z1 Z2 e2 , and m becomes the reduced mass of the nuclei involved. At
very low energies rc is large and so G rs =rc † ! 1. Thus
ÿ
!r
Z1 Z2 e2
2mc2
G E† E
»c 4"0
r
EG
; say;
ˆ
E
where
ÿ
2
EG ˆ 2mc
Z1 Z2 e2
»c 4"0 †
!2
and
E† 8.4
p
1
S 0†eÿ EG =E† :
E
8:7†
Doppler broadening of resonance peaks
We mentioned in }8.1 that the thermal motion of the nuclei in the target
affects the width of a resonance as seen experimentally. Neutrons in a
beam incident on a target, and mono-energetic with respect to that target,
are not mono-energetic with respect to the individual nuclei in the target,
112
Nuclear reactions
since these will be in random thermal motion. The energy that appears in
the Breit±Wigner formula is the energy in the centre-of-mass frame of the
neutron and the target nucleus. If the neutron has velocity v1 and the
nucleus velocity v2 , the centre-of-mass energy is
E ˆ 12m v1 ÿ v2 †2 ˆ E1 ‡
r
m
m
E2 ÿ 2
E1 E2 cos ;
M
M
where M is the mass of the nucleus, m ˆ mn M= mn ‡ M† is the reduced
mass, E1 is the centre-of-mass energy when thermal energy is neglected,
E2 is the thermal energy of the nucleus and the angle between v1 and v2 .
The term m=M†E2 can be neglected if the neutron energy is much
greater than thermal or if the target nucleus is heavy. The thermal energy
E2 is of order of magnitude kB T, where kB is Boltzmann's constant and T
is the temperature of the target. Since cos lies between ÿ1 and ‡1, it can
be seen that, when averaged over many nuclei, E will have a spread in
energy about E1 of magnitude
E 2
r
m
M
E 1 kB T :
Thus if a cross-section is measured in the laboratory as a function of E1 ,
in the neighbourhood of a resonance at energy E0 , the Breit±Wigner form
is modi®ed and, in particular, the width of the resonance peak will be
larger than the natural width by an amount of order
ÿ 2
r
m
E0 kB T :
M
This is Doppler broadening. A more detailed analysis shows that the total
area under the resonance peak is independent of temperature, so that the
height of the peak is reduced as the width increases. This is illustrated in
Fig. 8.5 for a resonance in 238 U. (We shall see in Chapter 9 that Doppler
broadening is of crucial importance for the thermal stability of nuclear
reactors.) In the resonance peak of Fig. 8.3, on the other hand, it is easy
to check that the effect of Doppler broadening at room temperature
Tr kB Tr 1=40† eV† is small.
Data on neutron scattering cross-sections will be found in McLane,
V., Dunford, C. L., and Rose, P. F. (1988), Neutron Cross Sections Vol. 2,
San Diego: Academic Press.
Problems
113
Fig. 8.5 The Doppler broadening of the Breit±Wigner cross-section ÿ for neutron radiative capture by 238 U. The resonance is at 6.67 eV and its natural width is
0.026 eV.
Problems
8.1 Quantum mechanics gives the total cross-section for scattering from an
impenetrable sphere of radius R at low energies kR 5 1† to be
ˆ 4R2 . For the cross-section of Fig. 8.2, show that the order of
magnitude of the cross-section between resonances is given by this formula with R the radius of the uranium nucleus, and at a resonance the
order of magnitude is given by =2†2 , where is the neutron wavelength ˆ 2=k†, as is implied by the Breit±Wigner formula (8.2).
8.2 Neutron detectors register individual neutrons by their production of
charged, ionising particles in a nuclear reaction. One method, appropriate to thermal neutrons E < 0:1 eV† uses the reaction
n ‡ 32 He ! p ‡ 31 H ‡ 0:73 MeV:
The cross-section for this reaction, which dominates at low energies,
follows the 1=v† law,
ˆ 0:039 c=v† b:
The mean distance a neutron travels through 3 He gas before it interacts
is l ˆ 1= He †, where He is the number density of helium atoms
(Appendix A). What detector thickness is needed, using 3 He gas at a
pressure of 10 bars (which gives He ˆ 2:4 1026 mÿ3 † in order that at
least 90% of incident neutrons with energy 0.1 eV produce ionisation?
114
Nuclear reactions
8.3 The nucleus 53 Li is apparent as a resonance in the elastic scattering of
protons from 42 He at a proton energy 2 MeV. The resonance has a
width of 0.5 MeV and spin 32.
(a) What is the lifetime of 53 Li?
(b) Estimate the cross-section at the resonance energy.
8.4 Figure 9.1(b) shows the measured total cross-section for neutrons incident on 238 U. What conclusion can you draw from the apparent absence
of 1=v† behaviour at low neutron energies?
8.5 The zero-temperature radiative capture cross-section illustrated in Fig.
8.5 is the intrinsic cross-section to which the Breit±Wigner formula is
immediately applicable. The excited state has spin 12, and there are two
signi®cant decay channels; the dominant one is ÿ-emission and the other
is neutron emission. Estimate the relative probability of neutron radiative capture at resonance, and estimate the elastic neutron scattering
cross-section at resonance. (Hint: use equation (D.11).)
9
Power from nuclear ®ssion
We saw in Chapter 4 that nuclei in the neighbourhood of 56 Fe have the
greatest binding energy per nucleon (Fig. 4.7). In principle therefore,
nuclear potential energy can be released into kinetic energy and made
available as heat by forming nuclei closer in mass to iron, either from
heavy nuclei by ®ssion or from light nuclei by fusion. This chapter is
devoted to the physics of nuclear ®ssion and its application in power
reactors. There were, world-wide, some 430 nuclear power stations operating in 1997, and these generated about 17% of the global electricity
supply. In the UK about 28% of all electricity generated came from
nuclear ®ssion.
9.1
Induced ®ssion
The spontaneous ®ssion of nuclei such as 236 U was discussed in }6.3; the
Coulomb barriers inhibiting spontaneous ®ssion are in the range 5±
6 MeV for nuclei with A 240. If a neutron of zero kinetic energy enters
a nucleus to form a compound nucleus, the compound nucleus will have
an excitation energy above its ground state equal to the neutron's binding
energy in that ground state. For example, a zero-energy neutron entering
235
U forms a state of 236 U with an excitation energy of 6.46 MeV. This
energy is above the ®ssion barrier, and the compound nucleus quickly
undergoes ®ssion, with ®ssion products similar to those found in the
spontaneous ®ssion of 236 U. To induce ®ssion in 238 U, on the other
hand, requires a neutron with a kinetic energy in excess of about
1.4 MeV. The `binding energy of the last neutron' in the nucleus 239 U
is only 4.78 MeV, and an excitation energy of this amount clearly lies
115
116
Power from nuclear fission
below the ®ssion threshold of 239 U. The differences in the binding energy
of the last neutron in even-A and odd-A nuclei are incorporated in the
semi-empirical mass formula in the pairing energy term (equation (4.5))
and are clearly evident in induced ®ssion. The odd-A nuclei
233
92 U;
235
92
U;
239
94
Pu;
241
94
Pu;
are examples of `®ssile' nuclei, i.e. nuclei whose ®ssion is induced even by
a zero energy neutron, whereas the even-A nuclei
232
90 Th;
238
92
U;
240
94
Pu;
242
94
Pu;
require an energetic neutron to induce ®ssion.
9.2
Neutron cross-sections for
235
U and
238
U
The principal isotopes of naturally occurring uranium are 235 U (0.72%)
and 238 U (99.27%). Figure 9.1 shows the total cross-sections tot and
®ssion cross-sections f of 235 U and 238 U for incident neutrons of energy
E from 0.01 eV to 10 MeV. Note that both scales on the graphs are
logarithmic. It is useful to divide the energy range into three parts and
pick out the features of particular interest. At very low energies, below
0.01 eV, the 1=v† law is clearly seen in the 235 U total and ®ssion crosssections, and the cross-sections are large, because of an excited state of
236
U lying just below E ˆ 0. The ®ssion fraction f =tot is 84%; the
remaining 16% of tot corresponds mostly to radiative capture (the formation of 236 U with ÿ-ray emission). In contrast, the cross-section for
238
U is very much smaller and nearly constant in this region, and is due
almost entirely to elastic scattering.
The second region is that between 1 eV and 1 keV, where resonances
are prominent in both isotopes. These resonances are very narrow and
radiative capture gives a signi®cant fraction of the total widths. This is
particularly true of the resonances in 238 U, which are below the ®ssion
threshold in this region; for example, ÿ-decays account for 95% of the
width of the resonance at 6.68 eV.
In the third region, between 1 keV and 3 MeV, the resonances are not
resolved in the measured cross-sections. Compound nuclear states at
these energies are more dense and wider. Thus the probability of radiative
capture is, on average, smaller than at lower energies. The ®ssion crosssection for 238 U appears above 1.4 MeV and the 235 U ®ssion fraction
9.2 Neutron cross-sections for
235
U and
238
U
117
Fig. 9.1 Total cross-section tot and ®ssion cross-section f as a function of
energy for neutrons incident on (a) 235 U, (b) 238 U. In the region of the dashed
lines the resonances are too close together for the experimental data to be displayed on the scale of the ®gures. Note that both the horizontal and vertical scales
are logarithmic. (Data from Garber, D. I. & Kinsey, R. R. (1976), Neutron Cross
Sections, vol. II, Upton, New York: Brookhaven National Laboratory.)
f =tot remains signi®cant. However, in both isotopes at these higher
energies the result of a neutron interaction is predominantly scattering,
either elastic scattering, or at higher energies, inelastic scattering with
neutron energy lost in exciting the nucleus. (The threshold energies for
118
Power from nuclear fission
inelastic scattering in 235 U and 238 U are 14 keV and 44 keV respectively,
which are the energies of the ®rst excited states in these nuclei.) Figure 9.1
shows that the 235 U and 238 U total cross-sections become similar, around
7 barns, at 3 MeV.
9.3
The ®ssion process
The measured widths of the low-energy resonances in the 235 U crosssection are 0:1 eV. The compound nuclei formed at these resonances
decay predominantly by ®ssion. Thus we can infer that ®ssion takes place
in a time of the order of
f ˆ
»
10ÿ14 s
ÿf
after neutron absorption (at least at low energies). On the time scales
relevant to this chapter we can regard this as instantaneous.
As with spontaneous ®ssion, there are generally two highly excited
®ssion fragments which quickly boil off neutrons. The average number of these prompt neutrons produced per ®ssion in 235 U is 2:5. The
value of depends somewhat on the energy of the incident neutron. In
addition there are on average d 0:02 delayed neutrons produced per
®ssion, emitted following chains of þÿ -decays of the neutron-rich ®ssion
products (}6.5). The mean delay time is about 13 s.
The total energy release on the induced ®ssion of a nucleus of 235 U is,
on average, 205 MeV and is distributed as shown in Table 9.1.
We have divided the energy release into that which becomes available
as heat, and that which is delayed by the long time scale of the þ-decay
chains of the ®ssion products. In the nuclear power industry the latter is
to some extent a nuisance. Some of it is delayed for decades or more and
presents a biological hazard in discarded nuclear waste. That which is
emitted during the lifetime of a fuel-rod is converted into useful heat, but
also presents a problem in reactor safety since there is no way of controlling it or turning it off, for example in the case of a breakdown in the heat
transport system. In the steady-state operation of a nuclear reactor we
shall see that ÿ 1† of the ®ssion neutrons must be absorbed in a non®ssion process somewhere in the reactor. Their radiative capture will yield
a further 3±12 MeV of useful energy in emitted ÿ-rays, which is not
included in the table. As for the neutrinos, their subsequent interaction
9.4 The chain reaction
119
Table 9.1. Distribution of energy release on the induced ®sion of a
nucleus of 235 U
MeV
Kinetic energy of ®ssion fragments
Kinetic energy of ®ssion neutrons
Energy of prompt ÿ-rays
Sub-total of `immediate' energy
Electrons from subsequent þ-decays
ÿ-rays following þ-decays
Sub-total of `delayed' energy
Neutrino energy
167
5
6
178
8
7
15
12
205
cross-sections are so small that almost all of their 12 MeV escapes unimpeded into outer space.
9.4
The chain reaction
Since neutron-induced ®ssion leads to neutron multiplication, in an
assembly of uranium atoms there is clearly the possibility of a chain
reaction, one ®ssion leading to another or perhaps several more.
Let us ®rst consider some of the length and time scales relevant to a
possible chain reaction in uranium metal, which we consider to be a
mixture of 235 U and 238 U atoms in the ratio c : 1 ÿ c†. The nuclear number density nuc of uranium metal is 4:8 1028 nuclei mÿ3 .
The average neutron total cross-section for a mixture of the two
isotopes is
235
238
‡ 1 ÿ c†tot
tot ˆ ctot
and the mean free path of a neutron in the mixture is
l ˆ 1=nuc tot
120
Power from nuclear fission
(cf. Appendix A). l is the mean distance a neutron travels between interactions. For example, the average energy of a prompt neutron from ®ssion is about 2 MeV, and at this energy we see from Fig. 9.1 that
235
238
tot
tot
7 barns. Thus l 3 cm. A 2 MeV neutron travels this distance in 1:5 10ÿ9 s.
The conceptually most simple case is that of an `atomic bomb' in
which the explosive is uranium highly enriched in 235 U. For simplicity
we take c ˆ 1, corresponding to pure 235 U. Figure 9.1 shows that a
2 MeV neutron has an 18% chance of inducing ®ssion in an interaction
with a 235 U nucleus. Otherwise, neglecting the small capture probability
at this energy, it will scatter from the nucleus losing some energy in the
process, so that the cross-section for a further reaction may be somewhat
increased. If the neutron is not lost from the surface of the metal, the
probable number of collisions before it induces ®ssion is about six
(Problem 9.4). Assuming thepneutron's
path is a `random walk', it will

move a net distance of about 6 3 cm 7 cm from its starting point, in
a mean time tp 10ÿ8 s, before inducing a further ®ssion and being
replaced by, on average, 2.5 new 2 MeV neutrons.
Not all neutrons will induce ®ssion. Some for example will escape
from the surface and some will undergo radiative capture. If the probability that a newly created neutron induces ®ssion is q then each neutron
will on average lead to the creation of q ÿ 1† additional neutrons in the
time tp . (We can neglect delayed neutrons in the present discussion.) If
there are n t† neutrons present at time t, then at time t ‡ t there will be
n t ‡ t† ˆ n t† ‡ q ÿ 1†n t† t=tp †:
In the limit of small t this gives
dn
q ÿ 1†
ˆ
n t†;
dt
tp
which has the solution
n t† ˆ n 0†e qÿ1†t=tp :
9:1†
The number increases or decreases exponentially, depending on whether
q > 1 or q < 1. For 235 U the number increases exponentially if
q > 1=† 0:4.
9.5 Nuclear fission reactors
121
Clearly, for a small piece of 235 U with linear dimensions much less
than 7 cm there will be a large chance of escape, q will be small, and the
chain reaction will damp out exponentially. However, a suf®ciently large
mass of uranium brought together at t ˆ 0 will have q > 0:4. There will
be neutrons present at t ˆ 0 arising from spontaneous ®ssion and, since
tp ˆ 10ÿ8 s, a devastating amount of energy will be released even in a
microsecond, before the material has time to disperse. For a bare sphere
of 235 U the critical radius at which q ˆ 1 is about 8.7 cm and the critical
mass is 52 kg. (See Problem 9.6.)
9.5
Nuclear ®ssion reactors
We now consider the fate of a 2 MeV neutron in a mass of natural
uranium c ˆ 0:0072†. It is possible for a 2 MeV neutron to induce ®ssion
235
238
and tot
are nearly equal at
in either of the two isotopes, but since tot
this energy, the neutron is much more likely to interact with 238 U since
this makes up more than 99% of natural uranium. In an interaction with
238
U, the probability of ®ssion is only about 5% of that of scattering,
which is the predominant interaction in this energy range. Because the
uranium nucleus is much more massive than a neutron, the neutron
would lose only a small proportion of its energy if it were to scatter
elastically (Problem 9.5(a)). However, a 2 MeV neutron is likely to scatter
inelastically, leaving the 238 U nucleus in an excited state, and after one or
two such scatterings the neutron's energy will lie below the threshold for
inducing ®ssion in 238 U.
Once its energy lies below the 238 U ®ssion threshold, the neutron has
to collide with a 235 U nucleus if it is to induce ®ssion. Its chances of doing
this are small unless and until it has `cooled down' to the very low energies, below 0.1 eV, where the 235 U cross-section is much larger than that
of 238 U (Fig. 9.1). In fact, before the neutron has lost so much energy it is
likely to have been captured into one of the 238 U resonances, and to have
formed the nucleus 239 U with the emission of ÿ-rays. In natural uranium
the proportion of ®ssion neutrons which induce further ®ssion is far too
small ever to sustain a chain reaction.
Basically, two routes have been followed to circumvent these dif®culties in producing a controlled chain reaction in uranium. The most highly
developed technology is that of thermal reactors, some of which are
fuelled by natural uranium. In a thermal reactor, uranium metal, or
more usually the ceramic uranium dioxide, is contained in an array of
fuel elements which are in the form of thin rods. Fission neutrons, while
122
Power from nuclear fission
still energetic, can escape from the rods into a surrounding large volume
®lled with material of low mass number and low neutron-absorption
cross-section, called the moderator. In the moderator the neutrons lose
their energy principally by elastic collisions (Problem 9.5(b)) and the
volume of the moderator is made suf®ciently large for a high proportion
of the neutrons to reach thermal energies corresponding to the ambient
temperature of the reactor 0:1 eV ˆ 1160 K†. These thermal neutrons, if
captured in the fuel rods, are predominantly captured by 235 U nuclei, the
large cross-section of 235 U at thermal energies compensating for its low
number-density. Since the neutrons slow down to thermal energies principally in the moderator rather than in the fuel rods, capture into the 238 U
resonances is largely avoided. The captures into 235 U lead to ®ssion with a
235
probability of f235 =tot
84% at thermal energies, and a chain reaction
may be sustained in the reactor in this way. The moderator used in
reactors fuelled by natural uranium is 12 C in the form of graphite, or
`heavy water', D2O.
The design criteria of thermal reactors are less stringent if the fuel is
arti®cially enriched with 235 U; the reactor can be made much smaller and
it becomes possible to use ordinary water rather than D2O as a moderator, despite the relatively high neutron-absorption cross-section of hydrogen through the reaction n ‡ p ! 2 H ‡ ÿ ‡ 2:33 MeV. Typical
enrichment in commercial reactors is 2%±3%.
The alternative to the thermal reactor is the fast reactor. In a fast
reactor a moderator is not required, and no large density of thermal
neutrons is established. Fission is induced by fast neutrons ± hence the
name. A fast reactor works because the ®ssion probabilities within the
fuel are increased over those of natural uranium by increasing the proportion of ®ssile nuclei to 20%. The ®ssile fuel used is 239 Pu rather than
235
U, for reasons we shall discuss in }9.7.
9.6
Reactor control and delayed neutrons
In a nuclear explosion the delayed neutrons are of no consequence: they
appear after the event. In a power reactor they must be considered, since
fuel rods can remain in the reactor for three or four years. Thus in a
reactor each ®ssion leads to ‰ ‡ d †q ÿ 1Š additional neutrons, where d
is the number of delayed neutrons per ®ssion.
In the steady operation of a reactor, with a constant rate of energy
production, the neutron density must remain constant so that the reaction
rate remains constant. Thus q must be such that the critical condition
9.6 Reactor control and delayed neutrons
123
‡ d †q ÿ 1 ˆ 0
is satis®ed.
Reactors are controlled by manipulating q mechanically, using adjustable control rods inserted in the reactor. The control rods contain materials such as boron or cadmium, which have a large neutron-absorption
cross-section in the thermal energy range (Fig. 8.3). Inserting or withdrawing the control rods decreases or increases q. It is important in the
design of reactors that the critical condition cannot be met by the prompt
neutrons alone, so that
q ÿ 1 < 0
always. Although the lifetime of a prompt neutron in a thermal reactor
may be as long as 10ÿ3 s, rather than 10ÿ8 s which we estimated in pure
235
U, this gives an uncomfortably short time scale in which to change q
mechanically and so avoid an accidental catastrophic exponential rise in
neutron density, as given by equation (9.1). However, since the reactor
can only become critical for
‡ d †q ÿ 1 ˆ 0
the time scale for a response to small variations in the population of
prompt neutrons is actually determined by the time scale of the delayed
neutrons, and becomes adequate for mechanical control. Problem 9.7
exempli®es this.
A reactor is brought into operation by slowly increasing q and allowing the neutron density to increase until the required power production
and operating temperature is reached. The heat produced, to be used in
the more traditional technology of raising steam and driving turbines, is
carried away by a coolant circulating through tubes which permeate the
core of the reactor, to a heat exchanger outside the reactor. Thus the
coolant is, necessarily, also a moderator, and its nuclear properties as
well as its thermal properties have to be considered. Gas-cooled thermal
reactors have commonly used carbon dioxide under pressure (typically 40
bar). Ordinary water can be used as coolant in reactors using enriched
uranium, such as pressurised-water reactors, in which the water is kept
under high pressure to prevent it boiling. In the case of a fast reactor, the
absence of moderator necessitates a highly compact core which demands
a coolant of high thermal conductivity and high thermal capacity; liquid
124
Power from nuclear fission
sodium appears to be most suitable and has been used in prototype
reactors.
For thermal stability, it is very important that q, the proportion of
neutrons inducing ®ssion, satis®es the inequality
dq
< 0;
dT
so that an increase in temperature T leads to a fall in q, and hence a fall in
the reaction rate and vice-versa. There are many factors affecting dq=dT,
arising from the thermal expansion of the various components of the
reactor, changes in the velocity distribution of the thermal neutrons
with temperature, and the effect of Doppler broadening of resonances.
In thermal reactors, Doppler broadening leads to an increase in the neutron absorption in 238 U resonances in the fuel rods and gives a signi®cant
negative contribution to dq=dT. Since the resonant cross-sections are
large, neutrons which impinge upon fuel rods and whose energies lie
near to resonances are absorbed close to the surface of the rod. The
broadening of the resonance increases the energy band absorbed and
hence increases the neutron absorption rate.
Parts of the No. 4 RBMK reactor at Chernobyl had dq=dT > 0 under
low power operation. This `design ¯aw' contributed to the catastrophic
accident in 1986. (All other RBMK type reactors have subsequently been
corrected.)
In a fast reactor the effects of Doppler broadening are more complicated since the ®ssion rate in 239 Pu resonances is also increased by broadening. It is important for the safety of fast reactors that the net effect on
dq=dT should be negative.
9.7
Production and use of plutonium
So far we have considered only 235 U as a nuclear fuel and regarded 238 U
with its high radiative-capture cross-section as something of a drawback.
However, the nucleus 239 U formed in radiative capture is odd and þdecays to the ®ssile nucleus 239 Pu:
239 ÿ
!239 Np
92 U
93
34 min†
‡ eÿ ‡ # 3:36 days†
239
94 Pu
‡ eÿ ‡ :
9.8 Radioactive waste
125
The nuclear properties of 239 Pu are very similar to 235 U and, in particular,
it is suitable as a fuel in a nuclear reactor. In a thermal reactor, some of
the 239 Pu produced will be burnt up in the lifetime of the fuel rods, and
the remainder may be extracted chemically from the spent fuel later.
Because of the relatively short mean life of plutonium isotopes (239 Pu
has a mean life for -decay of 3:5 104 years) virtually all plutonium on
Earth is man-made. Large quantities have been produced as a by-product
of the nuclear power industry (and wilfully for the nuclear weapons programme).
The value of for 239 Pu is 2.96 for fast neutrons, compared with 2.5
235
for U, so that it is a very suitable fuel for fast reactors. Such reactors
can be designed to breed more ®ssile 239 Pu from 238 U than is consumed,
using `spare' neutrons. In a fast reactor the central core is, typically,
loaded with 20% of 239 Pu and 80% of 238 U (`depleted' uranium recovered
from the operation of thermal reactors.) The core is enveloped in a
`blanket' of 238 U, and in this blanket more plutonium is made. A fastbreeder reactor programme can, in principle, be designed to utilise all the
energy content of natural uranium, rather than the 1% or so exploited in
thermal reactors.
Such schemes for burning plutonium in fast reactors have for the
most part been abandoned (}9.9), but 239 Pu can be burnt in thermal
reactors in the form of `MOX', a fuel of suitably mixed uranium and
plutonium oxides. Existing power plants designed for enriched uranium
fuel rods may need modi®cation before they can burn MOX: 239 Pu differs
from 235 U in having a lower fraction of delayed neutrons and a higher
neutron absorption cross-section, so that the use of MOX places more
stringent requirements on the control rods of the reactor.
9.8
Radioactive waste
The operation of a nuclear power programme generates radioactive
waste. After uranium and plutonium have been separated chemically
from the spent fuel, the remaining material, the `waste', consists mainly
of ®ssion products along with some higher actinides which have been
built up from uranium by successive neutron captures. The immediate
products of ®ssion are neutron rich, and hence þ-emitters. The daughter
nucleus from the þ-decay is often formed in an excited state, which then
decays to its ground state by ÿ-emission. þ-decay will then take place
again until the þ-stability valley is reached.
126
Power from nuclear fission
A complete description of the decay chains is well documented but
complex. Overall, for each ®ssion it is found that on average the rate of
release of ionising energy from the decay products at time t is given, to
within a factor of 2, for times between 1 s and 100 years by the formula
1:2
dE
1s
MeV sÿ1 :
ˆ 2:66
dt
t
9:2†
Over this period the energy release is divided roughly equally between
electrons and ÿ-rays. Energy lost to neutrinos is not included. Problem
9.8 indicates how such a simple empirical formula can be used to estimate
properties such as heat output and radioactivity of the waste.
The highly radioactive waste which remains after chemical processing
is kept in acid solution. The generally preferred option for the long-term
storage of this `high level' waste is vitri®cation, followed by deep burial.
Borosilicate glass, which readily dissolves large quantities of ®ssion products and actinides, has been used successfully for vitri®cation. Sites for
deep burial must have stable and suitable geological characteristics for at
least 10 000 years. In the UK, no site for deep burial has yet been found
acceptable to all the parties concerned, and liquid high level waste continues to be stored above ground in stainless steel tanks.
9.9
The future of nuclear power
The nuclear power scenario sketched out in }9.7 has not actually evolved.
Early economic forecasts of the cost of nuclear power were over-optimistic, and did not take properly into account the cost of decommissioning
power stations at the end of their working life, or the capital cost of
meeting increasingly stringent safety requirements. The nuclear explosion
at the Chernobyl power station in 1986 and its aftermath did nothing to
assuage an already existing public unease about nuclear power.
With fast reactors, there have been dif®cult engineering problems,
associated mostly with the hazardous use of liquid sodium as coolant.
All of the prototype fast reactors in the West have now been decommissioned. In fact, enough high-grade uranium ores have been discovered to
eliminate the need for expensive fast reactors for several decades. It is also
questionable if there is any need for spent fuel to be reprocessed, rather
than simply stored. However, fossil fuels and uranium ores will eventually
run out, and in a century or so fast reactors may be needed.
Problems
127
Except for France, in the West public hostility to nuclear power and
doubts about its economic viability have made the construction of new
nuclear power stations unlikely in the near future. Investment in nuclear
power continues in France and Japan, both countries having low reserves
of fossil fuels. China has ambitious plans for reactor building (despite
having an abundance of coal).
It is a great merit of nuclear power generation that it does not contribute to `greenhouse gases'. (See Problem 9.2.) This feature may well
become of compelling importance if global warming continues, and no
signi®cant progress is made in the use of non-fossil energy resources such
as hydroelectricity, wind power, and solar power, or in reducing the
energy demand in highly developed countries.
Problems
9.1 The combustion of methane
CH4 ‡ 2O2 ! CO2 ‡ 2H2 O;
releases an energy of about 9 eV/(methane molecule). Estimate the relative energy release per unit mass for nuclear (®ssion) as against chemical
fuels.
9.2 Show that a nuclear power plant producing 1000 MW of heat consumes
about 1 kg of 235 U (or other ®ssionable fuel) per day. Show that a power
station burning natural gas and producing 1000 MW of heat will discharge about 4000 tonnes of the greenhouse gas carbon dioxide into the
atmosphere every day.
9.3 Show that the semi-empirical mass formula predicts that for a heavy
nucleus the neutron separation energy (or `binding energy of the last
1
neutron') is approximately 2 11:2=A2 † MeV greater for an even Z
even N nucleus such as 236 U than it is for a nearby even Z odd N nucleus
such as 239 U.
9.4 Suppose that a neutron induces ®ssion in a nucleus with probability p,
and that otherwise the collision is elastic. Show that the mean number of
collisions it undergoes is 1=p.
9.5(a) A neutron with kinetic energy T0 (non-relativistic) collides elastically
with a stationary nucleus of mass M. In the centre-of-mass system the
scattering is isotropic. Show that on average the neutron energy after the
collision is
T1 ˆ
M 2 ‡ m2n
T0 :
M ‡ mn †2
128
Power from nuclear fission
(b) Consider the nuclei of a graphite moderator to be pure 12 C, with a
number density of 0:9 1029 nuclei/m3. For neutron energies less than
2 MeV the scattering is elastic, with a cross-section approximately constant 4:5 b.
Estimate (a) the number of collisions required to reduce the energy of
a 2 MeV ®ssion neutron to a thermal energy of 0.1 eV, and (b) the time
it takes.
9.6 If the neutron density r; t† in a material is slowly varying over distances long compared with the neutron mean free path l, r; t† approximately satis®es the `diffusion equation with multiplication',
@
ÿ 1†
ˆ
‡ Dr2 :
@t
tp
The coef®cient of diffusion is given in simple transport theory by
D ˆ lv=3, where v is the neutron velocity (assumed constant). At a
free surface, the effective boundary condition, again obtained from
transport theory, is
0:71l
@
‡ ˆ 0;
@n
where @=@n denotes differentiation along the outward normal to the
surface.
Using the data given in }9.4, estimate the critical radius of a bare
sphere of 235 U. Look for spherically symmetric solutions of the equation
of the form r; t† ˆ f r†et , and replace the boundary condition at the
surface r ˆ R by the approximation R ‡ 0:71l; t† ˆ 0.
9.7 In a simpli®ed model, the number of neutrons n t† in a reactor at time t is
given by
dn
q ÿ 1†
q
n‡ d
ˆ
dt
tp
tp
Z
0
n t 0 †eÿ tÿt †=þ 0
dt ;
þ
ÿ1
t
where tp is the mean life of the prompt neutrons and þ is the mean life of
those ®ssion fragments which produce delayed neutrons.
(a) Show how this equation may be derived, assuming that only one type of
®ssion fragment produces delayed neutrons.
(b) Show that solutions are of the form
n t† ˆ n0 et ,
and give the equation for .
Problems
129
(c) Show that if tp ˆ 10ÿ4 s, q ÿ 1 ˆ 10ÿ4 , and there are no delayed neutrons d ˆ 0), then n t† increases exponentially with a time scale of 1 s.
(d) Show that if þ ˆ 10 s, ‡ d †q ÿ 1 ˆ 10ÿ4 and q ÿ 1 ˆ ÿ0:0078 (corresponding to ˆ 2:5, d ˆ 0:02), n t† increases exponentially with a
time scale of about 13 minutes.
9.8(a) Show that the mean thermal power from a fuel rod of a reactor that has
been shut down for time t > 1 s†, after burning with steady power output P for a time T, is approximately
" #
1 s 0:2
1 s 0:2
ÿ
power ˆ 0:07P
:
t
T ‡t
(b) Before its catastrophic shut-down, the No. 4 Chernobyl reactor had been
producing about 3 GW of heat. Taking the mean age of its fuel rods to
be T ˆ 1 year, estimate the power outputs from the core at one week,
one month, and one year after the accident. 97% of the radioactive
material remained in the core.
9.9 The critical mass of a bare sphere of 239 Pu at atmospheric pressure is M,
say. By what factor must a bare sphere of mass 0:8M be compressed, for
it to become critical? (Assume that the critical radius of a sphere of
plutonium is proportional to the mean free path of a ®ssion neutron.)
10
Nuclear fusion
In this chapter we describe the nuclear reactions that power the Sun and
thus make possible life on Earth. In contrast to the power from ®ssion
discussed in the preceding chapter, the radiance of the Sun comes from
the fusion of the lightest element, hydrogen, into helium. We then examine the possibility of controlled nuclear fusion for power production on
Earth.
10.1
The Sun
In stars, the gravitational, the weak, the electromagnetic, and the strong
interactions all play an active and essential role. Our Sun and its planets
are thought to have condensed out of a diffuse mass of material, mostly
hydrogen and helium atoms, some 5 109 years ago. Table 10.1 gives the
estimated proportions of the ten most abundant nuclei in that mass of
material.
The major attributes of the Sun, determined from a wide variety of
observations, are as follows:
Mass
Mÿ ˆ 1:99 1030 kg
Radius
Rÿ ˆ 6:96 108 m
Luminosity
Lÿ ˆ 3:86 1026 W:
(The luminosity of a star is the total rate of emission of electromagnetic
energy.)
Because of the long range and universally attractive nature of gravity,
a homogeneous mass of gas at suf®ciently low temperature is unstable to
contraction into objects like stars. During contraction of a mass of gas,
130
10.1 The Sun
131
Table 10.1. The proportion by number, relative to carbon, of the ten most
abundant atoms in the Solar System at its birth
H
He
C
N
O
Ne
Mg
Si
S
Fe
2400
162
1.0
0.21
1.66
0.23
0.10
0.09
0.05
0.08
Data from Cameron, A. G. W. (1992), in Essays in Nuclear Astrophysics, ed. C. A.
Barnes, D. D. Clayton & D. N. Schramm. Cambridge University Press, p. 23.
gravitational potential energy is converted into kinetic energy and radiation energy, and the temperature of the gas rises. The rate of collapse is
determined by the extent to which the build-up of pressure in the hot,
dense interior can balance the incessant pressure of gravitational contraction. In a star like the Sun, as the temperature and density increased, its
rate of contraction was essentially stopped when the interior became hot
enough to ignite the hydrogen-burning reaction that we shall discuss in
detail presently. At this stage in the Sun's evolution, the generated nuclear
power keeps the interior hot enough to sustain the pressure that balances
gravity, and a quasi-static condition is established, a condition that exists
today. This condition is not one of thermodynamic equilibrium, since the
interior is hotter than the outside and the nuclear energy liberated at the
centre is transferred, radiatively and by conduction and convection, to the
surface, where it is radiated out into space, to our bene®t, and gives the
Sun its luminosity.
The principal reactions that power the Sun begin with the conversion
of hydrogen into deuterium:
p ‡ p ! 21 H ‡ e‡ ‡ e ‡ 0:42 MeV:
10:1†
This reaction involves the weak interaction (a proton changes to a neutron), and so occurs very rarely. It is the weak interaction that sets the
long time scale of the quasi-static state of the Sun.
The positron produced in the reaction quickly annihilates with an
electron to release a further 1.02 MeV of energy. The deuterium is converted to 3He:
p ‡ 21 H ! 32 He ‡ þ ‡ 5:49 MeV;
which in turn fuses to 4He:
10:2†
132
Nuclear fusion
3
2 He
‡ 32 He ! 42 He ‡ p ‡ p ‡ 12:86 MeV:
10:3†
Thus the net result of these reactions, which are called the `PPI chain', is
the conversion of hydrogen to helium with an energy release of
26.73 MeV per helium nucleus formed. The neutrinos emitted in the
p±p reactions take an average 0.26 MeV of energy each. This energy is
lost into outer space, but is not included in the observed luminosity. Thus
each hydrogen atom consumed in this process leads to the emission of
6.55 MeV of electromagnetic energy from the Sun.
The observed solar luminosity implies that Lþ = 6:55 MeV† ˆ 3:7 1038 hydrogen atoms are converted into helium per second. This rate of
conversion, over the lifetime of the Sun, gives a total of 5:4 1055 conversions. The Sun's mass and composition show that it started with about
8:9 1056 hydrogen atoms. We can conclude that less than 10% of the
hydrogen of the Sun has so far been consumed, and appreciate the long
time scale of this stage of stellar evolution.
Figure 10.1 shows the density, temperature and thermonuclear-power
density from a model calculation of the Sun as it is now. It it interesting to
note that 50% of the mass is within a distance of Rþ =4 from the centre,
and 95% of the luminosity is produced in the central region within a
distance of Rþ =5, where the temperature is such that kB T 0 1 keV. (kB
is Boltzmann's constant and kB T ˆ 1 keV when T ˆ 1:16 107 K.)
A simple order-of-magnitude calculation shows that the gravitational
energy released in contraction, before the quasi-static period began, is
suf®cient to produce such temperatures:
gravitational energy 2
GMþ
ˆ 3:8 1041 J;
Rþ
where the gravitational constant G ˆ 6:67 10ÿ11 m3 kgÿ1 sÿ2 . This
energy would give 1 keV of kinetic energy on average to every particle,
nuclei and electrons. At these temperatures the hydrogen and helium
atoms will be completely ionised; material in this condition is known as
plasma.
10.2
Cross-sections for hydrogen burning
We turn now to a more-detailed examination of the nuclear physics
involved in hydrogen burning. Reactions involving charged particles
were discussed in }8.3, where the role of the Coulomb barrier was empha-
10.2 Cross-sections for hydrogen burning
133
Fig. 10.1 (a) Mass densities and (b) the thermonuclear power density " and the
temperature T, in the modern Sun as a function of distance r from the centre.
(Taken from model calculations by Bahcall, J. N. et al. (1982), Rev. Mod. Phys.
54, 767.)
sised. For energies in the range of keV we shall use the low-energy expression for charged-particle reactions given by equation (8.7):
E† ˆ
where
p
1
S 0†eÿ EG =E†
E
10:4†
134
Nuclear fusion
ÿ
!2
Z1 Z2 e2
EG ˆ 2mc
:
»c 4"0 †
2
A more accurate expression may be necessary if there is a resonance in the
keV energy range.
Direct measurements of the cross-sections of reactions (10.2) and
(10.3) at energies of 1 keV have not been made since they are so small,
but values of S 0† are known by extrapolation from measurements at
higher energies. The p±d reaction (10.2) has been measured down to
15 keV and Spd 0† 2:5 10ÿ7 MeV b. The helium±helium reaction
(10.3) has been measured down to 33 keV, giving Shh 0† 4:7 MeV b.
The p±d cross-section is small because it necessarily involves a þ-transition to satisfy both energy and momentum conservation. The p±p reaction (10.1), the ®rst stage of hydrogen burning, has a cross-section which
is lower by many orders of magnitude because it involves the weak interaction. Although the reaction is crucial, the cross-section is so small that
it has not been directly measured in a laboratory at any energy.
Fortunately we have such a precise theory of the weak interaction that
the cross-section, and Spp 0†, can be calculated with some con®dence; it is
found that
Spp 0† 3:88 10ÿ25 MeV b:
10:5†
This order of magnitude is not too dif®cult to understand. As is explained
in Chapter 3, the proton±proton nuclear potential has been determined
from scattering experiments. From the potential, the low-energy proton±
proton nuclear-scattering cross-section can be calculated. Although there
is no resonance, the nuclear attraction makes the cross-section quite large,
36 barns, at energies 01 MeV which are low but are above the Coulomb
barrier. Reaction (10.1) involves the protons coming together within the
range of the nuclear force (Fig. 3.2) and, while they are together, a ÿdecay taking place. We can estimate the probability of this ÿ-decay by
probability ˆ
a typical nuclear time
:
a ÿ-decay time
Consider the cross-section at 1 MeV. Since the energy released in the
reaction is comparable with the energy released in the ÿ-decay of a free
neutron, it is reasonable to take the ÿ-decay time to be the neutron lifetime, 887 s, and the nuclear time is 1023 s (}5.2). The cross-section for
10.3 Nuclear reaction rates in a plasma
135
the proton±proton to deuteron reaction at 1 MeV should thus be of the
order
ÿ
!
10ÿ23 s
36 b† ˆ 4 10ÿ25 b:
887 s
Since the energy of 1 MeV is above the Coulomb barrier, we can infer
that Spp 4 10ÿ25 MeV b. This excellent agreement with equation
(10.5) is fortuitous, but the argument does make intelligible the order
of magnitude of this key quantity.
10.3
Nuclear reaction rates in a plasma
During the early cold stages of stellar contraction, nuclei do not have
kinetic energies high enough, compared with the Coulomb barriers
between them, for the barrier penetration probability to be signi®cant.
To obtain the reaction rate for a process in the interior of a star and see
how it depends on temperature, we must average suitably over the energies of the particles involved. The calculation is an important one so we
shall set it out here.
Consider the nuclei in a volume of plasma small enough for the
temperature and number densities to be considered uniform. We shall
assume that the velocities of the nuclei are given by the Maxwell±
Boltzmann distribution, so that the probability of two nuclei having a
relative velocity v in the range, v, v ‡ dv is given by
3
12 2
m 2 ÿmv2 =2kB T 2
e
v dv;
P v†dv ˆ
kB T
where m is the reduced mass of the pair. (The centre-of-mass motion has
been factored out.)
If the nuclei are labelled by a, b, with number densities a , b , the
number of reactions per unit time per unit volume is
reaction rate per unit volume ˆ Ka b vab †;
10:6†
where ab is the cross-section for the reaction, and the bar denotes the
average over the velocity distribution. Equation (10.6) follows from the
discussion of reaction rates in Appendix A. The factor K ˆ 1 if the nuclei
are different and K ˆ 12 if the nuclei are the same.
136
Nuclear fusion
We have
Z
vab ˆ
1
0
vab P v†dv:
Changing variables to E ˆ mv2 =2, and using the low-energy formula
(10.4) for the cross-section, this becomes
vab ˆ
8
m
12 1
kB T
32
Z
Sab 0†
0
1
eÿ E† dE;
10:7†
p
where E† ˆ E=kB T ‡ EG =E†.
The function eÿ E† is sharply peaked. It falls off rapidly at high
energies because of the Boltzmann factor, and at low energies because
of the barrier-penetration factor. The peak lies at E ˆ E0 where E† is a
minimum, i.e. where d=dE ˆ 0, which gives
1
2
E0 ˆ EG †3 kB T=2†3 :
Figure 10.2 is a graph of eÿ E† appropriate to the p±p reaction at the
centre of the Sun.
Fig. 10.2 The function exp ÿ E†† appearing in equation (10.7), plotted for the
proton±proton reaction at kB T ˆ 1:34 keV.
10.3 Nuclear reaction rates in a plasma
137
As it stands, the integral cannot be performed analytically, but the
main contribution comes from the peak. It is possible to replace E† in
the neighbourhood of E0 by a simpler expression, leading to an analytic
result for the integral (which is also a good approximation). The Taylor
expansion of E† about E0 gives
E† E0 † ‡ 12 E ÿ E0 †2 00 E0 †;
where
2
1
E0 † ˆ 3 12†3 EG =kB T†3 ;
1
ÿ1
5
00 E0 †; ˆ 3 12†3 EG 3 kB T†ÿ3 :
The linear term does not appear, since 0 E0 † ˆ 0. With this approximation, the integrand is replaced by a Gaussian peak and the integration
range can be extended down to E ˆ ÿ1 with negligible error. We have
R1
2
1
then, remembering the result ÿ1 eÿax dx ˆ =a†2 , the fairly simple
expression
vab ˆ
8
9Sab
0†
2
3mEG
12
2 eÿ
with
ÿ
mc2
ˆ 3 12† EG =kB T† ˆ 3
2kB T
2
3
1
3
!13 ÿ
!23
Za Zb e2
:
»c 4"0 †
10:8†
For practical calculations, taking the masses of nuclei to be A (one
atomic mass unit) gives
vab ˆ
7:21 10ÿ22 Aa ‡ Ab † Sab 0†
2 eÿ m3 sÿ1 ;
Za Zb
Aa Ab
1 MeV b
and
ÿ
Z2Z2 A A
ˆ 18:8 a b a b
A a ‡ Ab
!13 1
1 keV 3
:
kB T
138
Nuclear fusion
Note that the temperature dependence of v lies entirely in the factor
2 eÿ .
The temperature dependence is dramatic, as also is the dependence on
the nuclear species involved. Both are illustrated in Fig. 10.3, which
shows plots of 2 eÿ against temperature for several reactions of astrophysical interest. Note that the vertical scale extends over a range of 1060 !
For a given set of nuclear reactions, we can write down equations
giving the rate of change of the number densities of the nuclei participating, in a region of given temperature. For example, considering the
PPI reactions and writing ab ˆ vab , we have, from reactions (10.1) and
(10.2) and equation (10.6),
@d 1
ˆ 2pp 2p ÿ pd p d :
@t
Because of the long time scale of hydrogen burn-up, p may be regarded
as constant in this equation, giving the solution
d t† ˆ 12pp =pd †p 1 ‡ Ceÿpd p t †;
Fig. 10.3 The function 2 exp ÿ† appearing in equation (10.8), for some nuclear
reactions.
10.4 Other solar reactions
139
where C is a constant.
Using the numerical values for Spp 0† given in }10.2, and a temperature and density appropriate to the centre of the Sun, from Fig. 10.1, we
®nd that the time constant for establishing equilibrium is pd p †ÿ1
ˆ 3:3 s. Thus our assumption that p could be treated as a constant
was valid. In equilibrium, the ratio d =p ˆ 12pp =pd ˆ 1:5 10ÿ18 . The
low density of deuterium accounts for our neglect of d±d reactions (which
are considered in }10.6).
10.4
Other solar reactions
Our account above of hydrogen burning in the Sun is not complete. There
are other ways of consuming the 3He formed in reaction (10.2). The
presence of 4He in a star leads to the formation of 7Be:
3
2 He
‡ 42 He ! 74 Be ‡ þ ‡ 1:59 MeV:
10:9†
7
Be is unstable to the capture of a free electron from the plasma to form
Li:
7
7
4 Be
‡ eÿ ! 73 Li ‡ e ‡ 0:86 MeV;
10:10a†
or, with 10.3% probability, to form an excited state 73 Li :
7
4 Be
‡ eÿ ! 73 Li ‡ e ‡ 0:38 MeV;
10:10b†
which then decays:
7 3 Li
7
3 Li
! 73 Li ‡ þ ‡ 0:48 MeV:
is quickly broken up by a proton into two helium nuclei:
7
3 Li
‡ p ! 42 He ‡ 42 He ‡ 17:35 MeV:
10:11†
These reactions form the `PPII chain'.
Alternatively, the 74 Be may interact with a proton to form 85 B:
7
4 Be
8
5B
‡ p ! 85 B ‡ þ ‡ 0:14 MeV:
is unstable to ÿ-decay,
10:12†
140
Nuclear fusion
8
5B
! 84 Be ‡ e‡ ‡ e ‡ 14:02 MeV;
10:13†
and 84 Be breaks up into two helium nuclei:
8
4 Be
! 42 He ‡ 42 He ‡ 3:03 MeV;
10:14†
this is the `PPIII chain'. The positron annihilates with an electron to
release a further 1.02 MeV.
The relative importance of the PPII and PPIII chains, compared with
the PPI chain, can be calculated from the appropriate set of rate equations; in the standard model of the Sun, the PPI chain is the main process.
Another interesting set of reactions resulting in the burning of hydrogen to helium is the `CNO' cycle. The presence of any of the nuclei 126 C,
13
14
15
6 C, 7 N or 7 N catalyses the burning by the set of reactions
12
C ‡ p ! 13 N ‡ þ;
13
C ‡ p ! 14 N ‡ þ
14
N ‡ p ! 15 O ‡ þ;
15
N ‡ p ! 12 C ‡4He:
13
15
N ! 13 C ‡ e‡ ‡ e
O ! 15 N ‡ e‡ ‡ e
10:15†
The weak interactions in the cycle are not compelled to occur in a ¯eeting
10ÿ23 s, as in the p±p reaction, but can proceed at their leisure in the usual
ÿ-decay times. Carbon and nitrogen nuclei are known to be present in the
Sun (Table 10.1), but at the temperatures of the Sun the reaction rates are
greatly suppressed by the Coulomb barrier (Fig. 10.3), and the CNO cycle
probably accounts for only about 3% of stellar hydrogen burning. In
hotter stars the CNO cycle may dominate over the PP chains, since the
CNO cycle reaction rates increase more rapidly with temperature (Fig.
10.3 again).
10.5
Solar neutrinos
The solar reactions described in }10.1 and }10.4 lead to a considerable ¯ux
of neutrinos through the Earth. The ¯ux spectra predicted by the standard solar model are shown in Fig. 10.4. The band spectra result when the
neutrino is produced with an accompanying positron. For example, pp
refers to the process described by equation (10.1). Line spectra result
when there is no accompanying positron to share the energy release.
10.5 Solar neutrinos
141
Fig. 10.4 The solar neutrino spectra predicted by the standard solar model.
Spectra for the pp chain are shown by solid lines and those for the CNO chain
by dashed lines. (See Bahcall, J. N. and Ulrich, R. K. (1988), Rev. Mod. Phys. 60,
297.)
The line spectra marked 7Be come from the processes (10.10a) and
(10.10b). The pep line is from the three-body reaction
p ‡ eÿ ‡ p ! 21 H ‡ e ‡ 1:44 MeV:
10:16†
This is a very rare alternative to the process (10.1). Almost all of the
1.44 MeV released is taken by the neutrino and so does not serve to
heat the plasma. The hep spectrum comes from another very rare reaction, converting 32 He into 42 He:
p ‡ 32 He ! 42 He ‡ e‡ ‡ e ‡ 18:77 MeV:
10:17†
This is an alternative to the main mechanism given by equation (10.3).
The cross-sections for neutrino interactions are so small that solar
neutrinos arrive at the Earth more or less directly from the thermonuclear
furnace at the Sun's core. Measurements of the solar neutrino spectrum
provide a valuable check on our understanding of the Sun, and in fact the
measurements have also provided valuable information on the nature of
neutrinos themselves.
142
Nuclear fusion
Two basic techniques have been employed, that cover different
regions of the spectrum. The lowest energies are probed through the
neutrino reaction
71
ÿ
e ‡ 71
31 Ga ‡ 0:23 MeV ! 32 Ge ‡ e :
10:18†
As can be seen from Fig. 10.4, over 99% of all the solar neutrinos in the
energy range from the threshold energy for the reaction, 0.23 MeV, up to
a maximum energy of 0.42 MeV, come from the basic pp reaction (10.1).
Only the gallium experiments cover this low-energy region.
Although low-energy solar neutrinos are copious, their interaction
cross-section is particularly small (see }13.1). At the GALLEX experiment in Italy, 30 tons of gallium in a GaCl3±HCl solution serves as target.
The germanium produced binds chemically to form the volatile molecule
GeCl4, which is collected in the vapour above the liquid. The production
rate of germanium (a few molecules per day) is determined by observing
the characteristic Auger electrons and X-rays emitted in the K-capture
decays of 71Ge:
71
32 Ge
‡ eÿ ! 71
31 Ga ‡ e :
At the Homestake Mine experiment in the USA, the neutrino ¯ux is
measured through the reaction
ÿ
37
e ‡ 37
17 Cl ‡ 0:81 MeV ! e ‡ 18 Ar:
10:19†
The detector in this experiment consists of 615 tons of liquid perchloroethylene, C2Cl4. (The natural abundance of 37Cl is 24%.) The argon
produced in the experiment is extracted, and measured by monitoring
its decay by K-capture, in a similar way to the gallium experiment. The
threshold energy of 0.81 MeV makes the 37Cl detector blind to the neutrinos from the pp reaction. The detector is particularly sensitive to highenergy neutrinos from the 8B decay of equation (10.13).
An entirely different technique is used at the SuperKamiokande
detector in Japan. This looks for elastic scattering of solar neutrinos
from electrons in the target:
0
e ‡ eÿ ! e0 ‡ eÿ :
10.6 Fusion reactors
143
The target is 20 kilotonnes of very pure water, H2O. In many materials,
the atomic nuclei have large cross-sections for electron production
through reactions like (10.18) or (10.19). However, in the case of H2O
there is no such reaction for a proton, and the threshold energy for
electron production from 168 O (the principal 99.7% stable isotope of oxygen) is 14.9 MeV ± too high for all but a few solar neutrinos. Elastic
scattering from the electrons is thus the dominant reaction of solar neutrinos in water.
The water acts not only as target but also as detector. The scattered
electrons emit Cerenkov light, which is registered by photomultipliers. To
reduce background, counting is restricted to electrons with a recoil energy
greater than about 7 MeV. Hence the detector is sensitive only to 8B and
hep neutrinos.
The Cerenkov radiation gives information on the scattered electron's
direction, which is close to that of the incident neutrino. This directional
capacity makes water detectors more discriminating instruments than
gallium or chlorine detectors. Water detectors can also time individual
events. We shall see signi®cant applications of timing in }11.4.
For all neutrinos, event rates are necessarily very low, of the order of
one per day, and much patience is required to build up a signi®cant data
set. However, after several years accumulating data, all detectors tell a
consistent and signi®cant story: over the whole spectrum the event rate is
about one-half the expected rate. It is not thought that this discrepancy is
the result of our misunderstanding of the Sun, but that it is due to the
nature of neutrinos themselves. We reserve until Chapter 13 the explanations and implications of these observations.
10.6
Fusion reactors
For the generation of nuclear fusion power on Earth the immeasurably
slow p±p reaction is useless. However, Coulomb barriers for the deuteron,
2
1 H, are the same as for the proton, and the exothermic reactions
2
1H
2
1H
‡ 21 H ! 32 He ‡ n ‡ 3:27 MeV;
‡ 21 H ! 31 H ‡ p ‡ 4:03 MeV;
10:20†
suggest deuterium to be a suitable fuel for a fusion power station. The
natural abundance of deuterium is large, 0.015% of all hydrogen, and
144
Nuclear fusion
supplies of deuterium, in sea water for example, are effectively unlimited.
The mass ratio of 2:1 makes isotope separation relatively easy.
Current research is more concerned with deuterium±tritium mixtures
as fuel, using the reaction
2
1H
‡ 31 H ! 42 He ‡ n ‡ 17:62 MeV:
10:21†
This has two advantages over the reactions (10.20). First, the heat of
reaction is greater. Second, and more important, the cross-section is considerably larger (Fig. 10.5), because of an excited state of 52 He which gives
a resonance in the cross-section. The principal disadvantage is that tritium, 31 H, must be manufactured; it has no natural abundance since it
undergoes ÿ-decay with a mean life of 17.7 years. As Fig. 10.5 shows, the
peak of v is at kB T ˆ 60 keV, and a temperature of 20 keV is regarded
as a practical working temperature by fusion researchers.
A plasma at a temperature of 20 keV will vaporise any material container with which it comes into contact; current projects generally involve
pulsed devices which contain and heat the plasma for short bursts of time
only. For example, the moving electrically charged particles of the plasma
may be con®ned for short times, and even compressed, by magnetic ®elds,
Fig. 10.5 Values of v† (see equation (10.7) for the combined deuterium±deuterium reactions (10.20) and the deuterium±tritium reaction (10.21). (Data from
Keefe, D. (1982), Ann. Rev. Nucl. Part. Sci. 32, 391.)
10.6 Fusion reactors
145
and heated by electromagnetic ®elds. Instruments such as the Joint
European Torus (JET) at Culham are investigating these possibilities.
Inertial con®nement, by the implosion of small pellets containing the
deuterium±tritium fuel mixture, with the energy for implosion provided
by pulsed laser beams, is another active area of research. A continuing
series of `mini-explosions' of such pellets, each containing a few milligrams of fuel, is envisaged. The scenario for such a reactor usually
includes lithium in the heat-exchange blanket, since this provides a way
of breeding tritium through the reactions
7
Li ‡ n ‡ 2:46 MeV ! 3 H ‡
6
3
Li ‡ n ! H ‡
‡ n;
‡ 48 MeV:
(The natural abundances of 6Li and 7Li are 7.4% and 92.6% respectively.) The endothermic ®rst reaction can be brought about by the fast
neutrons produced in the deuterium±tritium reaction, and it is clear that
in principle a breeding ratio of greater than one is possible.
To achieve a temperature T in a deuterium±tritium plasma there must
be an energy input to the plasma of 4d 3kB T=2† per unit volume, where
d is the number density of deuterium ions and of tritium ions (i.e.
t ˆ d , and the electron density is 2d , giving 4d particles per unit
volume). The reaction rate in the plasma is 2d v. If the plasma is con®ned
for a time tc then, per unit volume of plasma,
fusion energy output 2d vtc 17:6 MeV†
ˆ
energy intput
6d kB T
10
ÿ19
3
10:22†
ÿ1
m s †d tc ;
evaluating the right-hand side at kB T ˆ 20 keV with the help of Fig. 10.5.
The plasma heating is certainly inef®cient, so that a substantial fraction of useful energy is lost in this process, and the conversion of fusion
energy to electricity is also (necessarily) inef®cient. Hence a requirement
for a useful device is that (fusion energy output)/(energy input) > 1, say.
From equation (10.22), this is equivalent to the criterion
d tc > 1019 mÿ3 s. This is known as the Lawson criterion. More stringent
formulations can be constructed for particular devices.
It should be appreciated that the engineering problems associated
with either magnetic or inertial con®nement as a basis for a working
power station are immense and have not so far been solved in practice.
146
Nuclear fusion
The Lawson criterion provides an estimate of how close a particular
design is to achieving practical results.
10.7
Muon-catalysed fusion
To end this chapter, we shall describe muon-catalysed fusion. This process
is experimentally well established, and quite well understood theoretically. The most interesting case, with power production in mind, is that
of muons incident on a dense mixture of deuterium D2 and tritium T2
molecules.
Muons result from the predominant mode of decay of negative pions
(see Fig. 3.5):
ÿ ! ‡ :
The mean life of a charged pion is 2:60 10ÿ8 s. The muon also decays
through the weak interaction (see }2.5):
! ‡ eÿ ‡ e ;
but it has a much longer mean life of 2:2 10ÿ6 s. Thus a beam of
negative pions is rapidly converted into a beam made up almost entirely
of muons. Pions are produced in nuclear reactions when particles accelerated to high energy interact with target nuclei. The energy cost of
producing muons by this route is estimated to be 5 GeV per muon
(very much greater than the rest mass energy of the muon 106 MeV).
The atomic and molecular physics of nuclei and muons is very similar
to that of nuclei and electrons, except for differences in scale. To a ®rst
approximation, binding energies scale by the mass ratio m =me † 207,
and distances scale by me =m † 1=207. The characteristic energy unit of
atomic and molecular physics, 1 Rydberg ˆ me e2 =4"0 †2 =2 »2 ˆ 13:6 eV,
is replaced by its muonic equivalent,
m e2 =4"0 †2 =2 »2 ˆ 2:81 keV:
The Bohr radius, a0 ˆ 4"0 † »2 =me e2 ˆ 0:529 A is replaced by the muonic Bohr radius,
a ˆ 4"0 † »2 =m e2 ˆ 256 fm:
10.7 Muon-catalysed fusion
147
In particular it is energetically highly favourable for muons to displace
electrons in atoms.
A muon incident on the D2±T2 mixture will lose energy mainly by
collisions with the electrons binding the D2 and T2 molecules (see Chapter
14), and when it has slowed suf®ciently it will break up a molecule and
take the place of an electron in either a deuterium or tritium atom.
Initially it is likely to be a highly excited state of this muonic atom, but
it will quickly cascade down to its ground state. Furthermore, muons
initially bound to deuterons in a d† atom will transfer to tritons in
subsequent collisions, since they are more tightly bound in the ground
state of a t† atom by 48 eV (Problem 2.9). The details of these processes
are complicated, but they are rapid, and happen in < 10ÿ8 s at liquid
hydrogen density.
The neutral t† atom is very small on the scale of a D2 molecule. It
can therefore move almost freely through the electronic cloud of a D2
molecule to join with a deuteron to form a dt†‡ ion. This ion is a
muonic analogue of the hydrogen molecular ion H2 †‡ . The process of
formation is very rapid compared with other possible competing processes, occurring in 5 10ÿ9 s. The reason for this is the existence of
a loosely bound (highly excited) state of the dt†‡ ion which, by happenstance, is almost in resonance with an excited vibrational state of the
composite dt†‡ ee d molecule. Thus the energy released by the formation of the dt†‡ ion can be transferred and dissipated.
The dt†‡ ion will fall to its ground state, losing energy to electrons.
In this ground state the distance between the d and t nuclei is reduced by a
factor 200, compared with the distance 1 A between the hydrogen
nuclei of a H‡
2 ion, and hence is 500 fm. Quantum tunnelling through
the Coulomb barrier allows the deuterion and triton to interact in
10ÿ12 s (see Problem 10.6):
d ‡ t ! 42 He ‡ n ‡ 17:62 MeV;
as in (10.21).
The nuclear energy released is taken up by the high recoil energies of
the -particle and neutron. The muon is usually freed, to repeat the cycle
(Fig. 10.6). However there is a small `sticking probability' !s that the
muon is captured by the positively charged -particle, and may remain
bound until it decays. Theoretical estimates give !s 0:8%. Though
small, this probability (rather than the mean life of the muon) is the
limiting factor in the number of fusions a muon can on average catalyse.
Problems
149
10.2 Why is the hydrogen content of the Earth so much less than that indicated in Table 10.1?
10.3 In the ÿ-decay of 8B, the neutrino takes on average about half of the
energy released. Estimate the contribution to the Sun's luminosity per
hydrogen atom consumed in the PPIII chain.
10.4 From Fig. 10.1, at the centre of the Sun kB T ˆ 1:35 keV and the mass
density of hydrogen is 5:6 104 kg mÿ3 .
(a) Using equations (10.5), (10.6) and (10.8) estimate the contribution to the
power density " from the PPI chain. Compare your result with Fig.
10.1(b).
(b) For the 12C±p reaction of the CNO cycle, S 0† ˆ 1:4 keV b. Estimate
the mean time it takes for a 12C nucleus at the centre of the Sun to be
converted to 13N.
10.5 A deuterium±tritium plasma contains d deuterium and d tritium nuclei
per unit volume.
(a) Show that to a good approximation d t† varies with time as
dd =dt ˆ ÿ2d v, where is the cross-section for the reaction
d ‡ t ! He ‡ n ‡ 17:62 MeV:
(b) If the plasma is brought together at time t ˆ 0 with d ˆ 0 , and con®ned for a time tc at constant temperature, show that the proportion of
the plasma `burnt up' is
v 0 tc †
:
1 ‡ v 0 tc †
(c) At kB T ˆ 20 keV, v ˆ 5 10ÿ22 m3 sÿ1 . What Lawson number 0 tc †
would be required to burn 5% of the fuel?
10.6 Take the SchroÈdinger equation for the relative motion of a deuteron and
triton bound by a muon to be
ÿ »2 =2M†r2 ý ‡ V r†ý ˆ Eý
where V r† ˆ
e2
md mt
‡ Kr2 , M ˆ
ˆ 1125 MeV=c2 ; M
4"0 r
md ‡ m t †
is the
reduced mass of the system. The term e2 = 4"0 r† gives the Coulomb
repulsion between the nuclei and the term Kr2 models the muon binding.
(a) Sketch V r†, and show that it has a minimum at r ˆ r0 where
r30 ˆ e2 =4"0 †=2K.
(b) Show that classically the system can make small oscillations about r ˆ r0
with frequency !, where
150
Nuclear fusion
!2 ˆ 3 e2 =4"0 †=Mr30 :
(c) What is the quantum energy of the system in its ground state, if
r0 ˆ 500 fm?
(d) Using the formula (6.15), suitably modi®ed, estimate the probability of
tunnelling through the Coulomb barrier at this energy.
(e) Hence estimate the mean life of the system.
11
Nucleosynthesis in stars
In the preceding chapter we explained how in a star like the Sun helium is
steadily formed from the fusion of hydrogen. In this chapter we sketch
some of the basic ideas of `nuclear astrophysics', a subject which seeks to
understand all the nuclear processes leading to energy generation in stars
in the various stages of stellar evolution, and to account for the observed
relative abundances of the elements in the Solar System in terms of these
processes.
The accepted theory of the Universe is that it is expanding, and began
with an intensely hot and dense `big bang' between 10 109 and 20 109
years ago. A few hundred thousand years after the big bang, the expanding material had cooled suf®ciently for it to condense into a gas made up
of hydrogen and helium atoms in a ratio of about 100:7 by number,
together with photons and neutrinos. Apart from a small amount of
lithium, it is thought that the proportion of heavier elements produced
in this ®rst explosion was insigni®cant (essentially because there are no
stable nuclei with A ˆ 5 or A ˆ 8). If this is so, we must conclude that all
the heavier nuclei in the Solar System have been produced in previous
generations of stars and then thrown out into space again, perhaps in the
explosion of supernovae.
11.1
Stellar evolution
Consider a star which has condensed from the primordial hydrogen±
helium mixture, and in which hydrogen burning has set in at the core.
As the hydrogen in the core is consumed, the reaction rate eventually
becomes insuf®cient to sustain the temperature, and hence the pressure,
151
152
Nucleosynthesis in stars
that prevents further gravitational contraction. Thus more material falls
into the core region. If the star is massive enough, the gravitational energy
released raises the temperature of the core suf®ciently for helium to begin
burning at a signi®cant rate. As the helium in turn is consumed, further
stages of nuclear burning set in until the most tightly bound elements,
iron and nickel, are formed. At each stage a higher temperature is needed
to overcome a higher Coulomb barrier; the energy for this is provided by
gravitational contraction.
Before considering these later stages of nuclear burning in more
detail, it is important to appreciate that there are conditions under
which the central pressure can permanently balance the pressure exerted
by gravity. Then contraction will cease and the temperatures for further
steps in nucleosynthesis will not be reached. After completing as much
burning as it can, the star will simply cool. The ®rst contribution to the
pressure that may stop contraction is the `electron-degeneracy pressure'.
Since electrons are fermions, it follows from the Pauli principle that, even
in a cold star with T 0 K, electron states are occupied up to an energy
"F ˆ »2 =2me †k2F , where (Appendix B, equation (B.5)) k3F ˆ 32 e , and e
is the number density of electrons. Thus in matter at high density there
exist electrons with high kF and hence high kinetic energy, which necessarily exert a high pressure. To obtain a simple order-of-magnitude estimate of this effect, we set the density of matter in a star of mass M, radius
R, to be constant. Then the number of electrons in the star is
Ne ˆ M=†, where is the stellar mass per electron. For material with
Z ˆ N, we have ˆ 2 amu, to a good approximation. The electron
number density is e ˆ M=†= 4R3 =3†, giving
k3F ˆ 32 e ˆ
9 M 1
:
4
R3
11:1†
Assuming that the electrons can be treated non-relativistically, the total
kinetic energy of Ne electrons at T ˆ 0 is 3=5†Ne "F (cf. Problem 5.2). At
T 0 K, the sum of the electron kinetic energies and the gravitational
potential energy is therefore
2
3 M »2 9M 3 1
3 GM 2
ÿ
:
Eˆ
5 2me 4 R2 5 R
11:2†
The star begins its life with R large, and the electron energy is then
much smaller than the gravitational energy. As the star evolves it con-
11.1 Stellar evolution
153
tracts, so that our `model' R decreases and E becomes more negative. The
energy released goes into heating the interior of the star and into radiation. However, no more energy can be released by contraction when E
reaches its minimum value where dE=dR ˆ 0, at
Rmin
23
9
»2
ˆ
4 Gme M 13 53
1
Mþ 3
ˆ 7:2
103 km; taking ˆ 2 amu:
M
11:3†
(A calculation which does not make our assumption of constant density,
but determines the density self-consistently, gives a numerical coef®cient
of 8.8 instead of 7.2.)
The corresponding mass density in our model at this minimum radius
is
M
4M 2 G3 m3e 5
ˆ
3
4Rmin =3†
273 »6
M 2
ˆ 1:27
109 kg mÿ3 :
Mþ
mass ˆ
11:4†
There are many stars with masses similar to, but generally smaller than,
the Sun which are close to this inert condition. They have high density
and small radii, and are called white dwarfs. It may be noted that the
minimum radius decreases as the mass increases.
The maximum electron momentum in our model when R ˆ Rmin is,
using equation (11.1),
M
pF ˆ »kF ˆ 3 † mass =† » ˆ 0:44
Mþ
2
1
3
1
3
23
MeV=c:
Since the rest mass of an electron is 0.511 MeV/c2 , the assumption in our
calculation above that the electron can be treated non-relativistically, i.e.
that pF 5 me c, is clearly suspect for stars with M Mþ and is certainly
wrong for stars of large M. In the limit when M is large, we take
1
" ˆ p2 c2 ‡ m2e c4 †2 pc ˆ »ck;
154
Nucleosynthesis in stars
for all the electrons, so that the total energy of Ne electrons at T ˆ 0
becomes 3=4†Ne »ckF † (Problem 11.1). Hence, using equation (11.1)
again, the expression (11.2) for the energy is replaced by
"
#
13 43
3
9
M
3
2 1
ÿ GM
E
»c
:
4
4
5
R
11:5†
If M is suf®ciently large the coef®cient of 1=R† is negative and there
is no minimum energy: electron degeneracy alone cannot prevent the
collapse of the star. Our extreme relativistic approximation becomes
increasingly valid as R decreases. Equation (11.5) suggests that the critical
value of M is
Mˆ
3
1 15 5†2 »c 2
ˆ 1:74 Mþ :
16 2 G
A more careful calculation takes proper account of relativistic energies and determines the density distribution self-consistently. It is then
found that the electron-degeneracy pressure cannot stop the gravitational
collapse of a star of mass M if M > 1:44 Mþ . This result, due to
Chandrasekhar, is known as the Chandrasekhar limit.
At very high densities of matter it becomes energetically favourable
for electrons to be captured by protons, and a Fermi gas of neutrons is
formed. Thus ®nal collapse may be prevented by neutron-degeneracy
pressure. The number of neutrons in such a neutron star is approximately
(M=1 amu). Putting ˆ 1 amu and replacing me by mn in (11.3) and
(11.4) suggests a neutron star has a radius of
R ˆ 1:26 Mþ =M†1=3 10 km;
11:6†
and a corresponding mass density 2.37 M=Mþ †2 1017 kg mÿ3 . Such a
mass density is comparable with the mass density of nuclear matter, so
that our simple expressions, which neglect nuclear interactions, are at best
only order of magnitude estimates. Nevertheless, we expect there to be a
mass limit, analogous to the Chandrasekhar limit, beyond which neutrondegeneracy pressure cannot stop further gravitational collapse. Putting
ˆ 1 amu in (11.6) suggests this limit is about four times the
Chandrasekhar limit. More realistic calculations, taking into account
the compressibility of nuclear matter, give the maximum possible mass
11.2 From helium to silicon
155
of a neutron star to be 3Mþ . Neutron stars having a greater mass than
this will collapse. Newtonian gravitation theory becomes inadequate to
describe what will happen at very high mass density. According to the
general theory of relativity, the star will collapse into a black hole, manifested only by its intense gravitational ®eld.
11.2
From helium to silicon
We return now to the problem of nucleosynthesis beyond helium. It is
clear that the fusion of hydrogen to helium already converts most of the
available nuclear potential energy into heat and radiation. The binding
energy per nucleon in 4He is 7.1 MeV, and there is only a further 1.7 MeV
per nucleon to be released in complete burning to iron. Also, as can be
seen from Fig. 10.3, as the elements involved become heavier and more
charged, higher and higher temperatures are required for there to be
signi®cant tunnelling through the Coulomb barriers. In fact, as we shall
see, the simple fusion process is superseded by another when elements
around 28
14 Si have been produced.
A few of the important reactions associated with helium burning to
oxygen, and oxygen to silicon, are listed below, along with typical temperatures and mass densities at which in a suf®ciently massive star they
are calculated to occur:
4
He ‡ 4 He ! 8 Be
4
He ‡ 8 Be ! 12 C ‡ ÿ ‡ 12 0:61 MeV†
4
He ‡ 12 C ! 16 O ‡ ÿ ‡ 16 0:45 MeV†
16
)
kB T 10--20 keV†
105 --108 † kg mÿ3
4
O ‡ 16 O ! 28
14 Si ‡ He ‡ 32 0:30 MeV† kB T 100--200† keV
109 kg mÿ3 :
The initial stage of helium burning needs some explanation. As Table
4.2 indicated, 4He has the largest binding energy per nucleon of any
nucleus less massive than 12C. The most stable form of nuclear material
with A < 12 is therefore 4He, and in particular 84 Be does not exist as a
stable nucleus. Nevertheless 84 Be exists as a resonant state that is seen in
the laboratory in ± scattering at an energy of 94 keV, in the centre-ofmass frame, with a narrow width (due to the Coulomb barrier) of 2.5 eV.
In a 4He plasma this state is established with an equilibrium density such
that the rate of production equals the rate of decay. Thus the `mass gaps'
156
Nucleosynthesis in stars
at A ˆ 5 and A ˆ 8 can be bridged. The next step in the chain,
8
Be ‡ ! 12 C ‡ ÿ, is in fact enhanced because it is a resonant reaction.
There is an excited state of 12 C at 0.29 MeV above the 8 Be ‡ threshold.
In the ®nal stages of oxygen burning, core temperatures in the star are
calculated to reach 300±400 keV, with mass densities in excess of
109 kg mÿ3 .
11.3
Silicon burning
In all the preceding stages of stellar evolution, photons have always been
present in thermal equilibrium with the plasma. They have played an
important role in radiative heat transfer, but have been unimportant
for initiating the nuclear processes we have discussed. However, a photon
couples electromagnetically to a nucleus and can be readily absorbed by a
nucleus to form an excited state. If the photon has an energy above the
threshold for nuclear break-up of that nucleus, break-up can occur. This
process is called photodisintegration.
As the temperature in the core of a star approaches kB T ˆ 1 MeV,
the increasing number of photons in the high-energy tail of the thermal
distribution makes photodisintegration an important process. In particular, protons, neutrons and -particles are knocked out of nuclei.
Although this effectively undoes some of the nuclear binding that has
gone on before, protons and -particles, as well as neutrons, are at
these temperatures readily accepted into any nucleus present, and a situation approaching thermal equilibrium is quickly established with the most
tightly bound elements, iron and nickel, copiously produced.
At this stage the core of a massive star is in an unstable condition.
There is no more nuclear fuel to burn to delay further gravitational contraction, so even higher densities and temperatures occur. It then becomes
energetically advantageous for electrons at the top of the Fermi distribution to undergo electron capture to form neutron-rich nuclei, which on
Earth would be þÿ -unstable. This process removes heat from the core by
producing neutrinos which escape, as well as removing electrons. Thus
the pressure falls, hastening contraction and leading to the removal of
even more electrons. Eventually there will be a catastrophic collapse of
the core, an implosion which can only be stopped by nucleon pressure
and the nucleon±nucleon short-range repulsion.
The cooler regions of the star outside the core will contain unburned
or only partially burned material. As the core implodes, these regions will
quickly fall inwards and rise in temperature so that the remaining fuel
11.4 Supernovae
157
burns explosively, blowing the stellar outer mantle into space. This is the
scenario for neutron star formation accompanied by a supernova explosion.
11.4
Supernovae
In a supernova, it is only the core of the star that collapses to form a
neutron star or a black hole. An important feature of core collapse is the
large gravitational energy release. We may understand the order of magnitude of the energies involved using the simple model described in }11.1,
adapted to neutron degeneracy. If the core has collapsed to radius R, and
radiated all of the heat generated, its energy in the model is
E R† ˆ
A B
ÿ ;
R2 R
3 9 2=3 »2 M 5=3
3
where A ˆ
, B ˆ GM 2 , and M is the mass of the
2mn mn
5 4
5
core.
In equilibrium, the core will adopt the radius Rmin ˆ 2A=B, which
minimises this energy. The energy taken in compressing the neutrons is
A=R2min . Since
A
B2
B
ˆ
ˆ
;
2
Rmin 4A 2Rmin
this energy is only half the gravitational energy released. (Note that the
initial energies are negligible.) Almost all of the other half of the energy
goes into heating the core.
As an example, we shall consider a core with a mass of 2Mþ . This
core mass is above the Chandrasekhar limit. The core will contain N ˆ
2Mþ =mn † ˆ 2:4 1057 neutrons, and Rmin ˆ 10:0 km, using equation
(11.6). The heat energy per neutron is
B
ˆ 83 MeV:
2Rmin N
Such an energy release is almost 10% of the rest-mass energy of a
nucleon, and about ten times the energy per nucleon released in the
whole of the star's previous history (see }11.2)!
158
Nucleosynthesis in stars
After silicon burning, and at the start of collapse, about one-half of
the nucleons in the core are protons. The reaction
eÿ ‡ p ! n ‡ e ;
which takes place during collapse, will therefore produce N=2 ˆ 1:2 1057 electron neutrinos. Detailed calculations show that collapse takes
place in 10ÿ3 s, and the average neutrino energy is about 10 MeV.
Thus these neutrinos carry away about 6% of the gravitational energy
released: the rest is left as heat in the collapsed core.
With heat energies of 80 MeV per neutron, and densities approaching that of nuclear matter, a transient state of thermal equilibrium will
exist, containing not only neutrons, protons, and electrons, but also electron±positron pairs, photons, neutrinos and anti-neutrinos coupled in by
reactions such as
e‡ ‡ n $ p ‡ e ;
eÿ ‡ p $ n ‡ e :
Muon and tau neutrinos and anti-neutrinos will also be present, induced
by reactions such as
e‡ ‡ eÿ $ ‡ ;
e‡ ‡ eÿ $ ‡ ;
which proceed through the intermediary of the neutral Z boson.
For thermal equilibrium to be established, the mean free path of the
particles participating in equilibrium must be less than the size of the
system. Cross-sections for neutrino scattering will be discussed in
Chapter 13. At the density of nuclear matter this condition is easily
satis®ed (Problem 13.3), so that neutrinos and anti-neutrinos take part
in thermal equilibrium. On the other hand, their mean free path is very
large compared with that of photons or electrons, so that neutrinos and
anti-neutrinos take the place of photons in transferring heat from the
interior to the surface of the core, and radiating it away.
The time scale for this neutrino cooling of the core is of the greatest
interest. Detailed calculations suggest a neutrino will diffuse from the
core to the surface in 1 s (Problem 13.3), and that appreciable cooling
will take place in 10 s. A supernova is an impressive event: the star
emits ten times as much energy in 10 s as it has previously emitted in
billions of years of stellar evolution. Assuming an average energy of
10 MeV for neutrinos and anti-neutrinos, the total number of all types
11.4 Supernovae
159
emitted must be 80N MeV=10 MeV 2 1058 . This number greatly
exceeds the number of electron neutrinos squeezed out in the initial collapse of the core.
Strong support for the supernova scenario we have outlined, and in
particular the role played by neutrinos, was provided by the observations
of a burst of neutrinos accompanying the 1987 supernova explosion in the
Large Magellanic Cloud. The LMC is a nearby galaxy at a distance D of
about 1:5 1021 m from the Solar System. This was the ®rst, and so far is
the only, local supernova explosion to have occurred while neutrino
detectors have been in place. If in total N neutrinos and anti-neutrinos
were emitted, we should expect N =4D2 ˆ N =1057 †3:5 1013 mÿ2 to
arrive at Earth over a few seconds. Of the instrument types described
in }10.5, the water detector is most appropriate, since it gives the vital
information on the arrival times of those neutrinos and anti-neutrinos it
detects.
Note also that water detectors are sensitive to energetic electron antineutrinos e , through the reaction
e ‡ p ! n ‡ e‡ ;
on the protons in the water. The cross-section for this reaction is of
similar size to the cross-sections of neutrinos in the gallium and chlorine
detectors, about two orders of magnitude greater than the cross-section
for neutrino±electron scattering.
Taking N 1058 , gives an expected integrated ¯ux of 1014 mÿ2
supernova neutrinos and anti-neutrinos over a few seconds. For comparison, the ¯ux of 8B and hep neutrinos calculated using the standard solar
model is 6 1010 mÿ2 sÿ1 . These are the neutrinos to which the
Kamiokande II detector was sensitive, and in the typical working conditions of the detector gave an event rate of about one per day.
An observed burst of twelve events within 13 s, and with energies
between 6 and 36 MeV, was clearly high above any background of
solar neutrino events, and correlated with the optically observed supernova explosion. The IMB detector in the USA, which is another water
detector, also observed six events within 6 s, with energies between
20 MeV (the detector threshold) and 40 MeV.
Analysis of the data from the two detectors suggests that the surface
temperature T of the source was of the form
T 4 ˆ T04 eÿt= ;
160
Nucleosynthesis in stars
with T0 ˆ 4:2 MeV and ˆ 4:6 s, and the integrated ¯ux of anti-neutrinos was 1:3 1014 mÿ2 . These measurements are entirely consistent with
the picture of stellar collapse we have outlined, and we can have some
con®dence in identifying the source as a supernova.
11.5
Nucleosynthesis of heavy elements
The most likely process for the formation of elements heavier than those
grouped around iron, produced in the silicon burning described above, is
neutron capture. If a supply of free neutrons is available, they can accrete
on an iron-group seed nucleus by radiative capture, unimpeded by
Coulomb barriers, to build up a neutron-rich isotope. As the neutron
number in the nucleus increases it will become unstable to þÿ -decay,
thus forming a new element of atomic number Z ‡ 1 from an element
of atomic number Z. Successive neutron captures, interspersed with þÿ decays, can eventually build up many, but not all, of the heavy stable
nuclei. Since the build-up follows the neutron-rich side of the `þ stability
valley' (}4.6), some of the proton-rich stable isotopes are inaccessible in
this process. It is an interesting fact that such isotopes have a much
smaller natural abundance than their neutron-rich neighbours.
There are two basic time scales in this scenario of heavy element
synthesis by neutron accretion: the þ-decay lifetimes and the time intervals between successive neutron captures (which are inversely proportional to the capture cross-sections and the neutron ¯ux). If the rate of
neutron capture is slow compared with the relevant þ-decay rates (the sprocess) the nuclei that are built up will follow the bottom of the þ
stability valley very closely. If the rate of neutron capture is rapid (the
r-process) highly unstable neutron-rich isotopes will be formed which
cascade down to stable nuclei, some of which are inaccessible by the sprocess; thorium and uranium must have been formed in this way. The
observed nuclear abundances, especially in the regions of closed-shell
nuclei, suggest that both the r- and the s-processes have played a part
in the synthesis of nuclei found in the Solar System and, in particular, the
heavy elements found on Earth.
The site of the r-process is believed to be in supernovae explosions
close to the region of neutron star formation, where over a short period of
time large neutron ¯uxes can be expected. The s-process probably occurs
during helium burning in massive stars, where a low neutron ¯ux can be
provided by a number of reactions, for example (Fig. 8.4),
Problems
161
‡ 136 C ! 168 O ‡ n:
For nucleosynthesis of the heavy elements by the s-process there must be
iron present, derived from nucleosynthesis in previous generations of
stars and forming part of the gas from which the star in question condensed.
In this chapter we have attempted to provide no more than a qualitative sketch of a theory which is still being developed. Many of the basic
components of the theory are probably in place but important aspects are
still being investigated through laboratory measurements and theoretical
estimates of reaction rates, and computer studies of reaction networks,
combined with stellar models. A rich variety of facts and phenomena
remains to be explained.
Problems
11.1 In a plasma with high electron number density e , using the extreme
relativistic approximation in which energy and momentum are related by
E ˆ pc, show that the average energy of an electron is 3=4† »ckF , where
kF is given by k3F ˆ 32 e .
11.2 The planet Jupiter is composed mostly of hydrogen. It has mass
1:9 1027 kg and mean radius 7 107 m. Show that if it were uniformly dense its gravitational energy per particle would be only 7 eV,
too small to ignite nuclear reactions.
11.3 Estimate the mass density of (metallic) hydrogen at 0 K at which it is
energetically favourable to subtract electrons from the electron gas and
form neutrons by the inverse þ-decay
p ‡ eÿ ! n ‡ :
11.4 Estimate the ratio of the number of protons and electrons to the number
of neutrons in a neutron star at the density of nuclear matter, in thermal
equilibrium at low temperature.
11.5 The Planck radiation law states that the number of photons per unit
volume in an energy range dE is
1
E 2 dE
:
2 »c†3 eE=kB T ÿ 1
At a temperature kB T ˆ 500 keV, estimate the number of photons per
unit volume with an energy greater than 8 MeV.
11.6 The cross-section for 8Be production in ± scattering at energy E in the
centre-of-mass frame is given by the Breit±Wigner formula
162
Nucleosynthesis in stars
E† ˆ
2 »2
ÿ2
;
m E E ÿ E0 †2 ‡ ÿ2 =4
with E0 ˆ 94 keV, ÿ ˆ 2:5 eV. (Note the additional factor of 2 in the
Breit±Wigner formula for identical particles: see Problem D.1.) In a
plasma at temperatures kB T 4 ÿ, the thermal energy v is dominated
by energies in the neighbourhood of E0 .
(a) Show that
v 16 »2 ÿ
m kB T
32
eÿE0 =kB T :
(b) Hence show that the density Be of 8Be in equilibrium with -particles of
number density is in the ratio
ÿ
!32
Be »
4 »2
ˆ
v ˆ eÿE0 =kB T :
2 ÿ
m kB T
(c) Calculate this ratio for kB T ˆ 15 keV and a helium mass density of
106 kg mÿ3 .
12
Beta decay and gamma decay
In this chapter we present some of the theory of þ-decay and ÿ-decay. In
both cases, a fuller treatment requires more quantum mechanics than is
usually contained in an undergraduate course, but we shall see that much
of the experimental phenomena can be understood qualitatively without
the complete relativistic theory.
12.1
What must a theory of þ-decay explain?
In þ-decay, introduced in }3.5, the charge of a nucleus changes while A
remains ®xed. This occurs either by the simultaneous emission of an
electron and an anti-neutrino, or a positron and a neutrino, or by the
capture of an atomic electron with the emission of a neutrino. The appropriate stability conditions were discussed in }4.6. Several nuclei, for example 64Cu, can decay by any of these processes (Fig. 4.5). In electron
capture, the neutrino energy and the recoil energy of the nucleus are
sharply de®ned. In the other processes, the electron (or positron) can
take any energy between zero and the maximum allowed by energy conservation. Figure 12.1 shows the experimentally determined energy spectra for electron emission and for positron emission from 64
29 Cu. It was the
observation of continuous energy distributions such as these that led
Pauli to infer the existence of the neutrino in 1931: given that the energy
levels of a nucleus are discrete, the electron and nuclear recoil energies in
the centre-of-mass frame would by energy and momentum conservation
be likewise discrete, unless some third particle (the neutrino) were present
to share energy and momentum. The neutrino mass can be deduced to be
small since, when the electron takes its maximum energy, the energy
163
164
Beta decay and gamma decay
Fig. 12.1 Electron and positron energy spectra from the þ-decay of 64Cu, giving
the probability distributions in energy (both normalised to unity) from a large
sample of decays. The experimental points are from Langer, L. M. et al. (1949),
Phys. Rev. 76, 1725. The curves are ®ts to the data using equations (12.5) and
(12.6).
balance to within the accuracy of present experiments is complete.
Maximum electron energy corresponds to the neutrino carrying no
momentum, so that the neutrino energy then would be its rest-mass
energy m c2 . We consider recent experimental limits on m in }13.2.
Because the neutrino interacts so weakly with other particles, it was not
12.1 What must a theory of þ-decay explain?
165
until 1959 that its existence was more directly con®rmed by the observation of the reaction
e ‡ p ! n ‡ e‡ ;
using the high anti-neutrino ¯ux associated with a nuclear reactor (which
arises from þ-decays of the neutron-rich ®ssion fragments).
As in the case of ÿ-decay mean lives, þ-decay mean lives span many
orders of magnitude. For example, the most common isotope of indium,
115
14
49 In, is þ-unstable, but its mean life is 10 years, whereas þ-decay mean
lives of the order of seconds or minutes are common. As with ÿ-decay,
mean lives depend strongly on the nuclear-spin change in the decay.
The ®rst experimental evidence for the violation of mirror symmetry
at the subatomic level was found in þ-decay by Wu in 1957, following a
suggestion by Lee and Yang. The experiment measured the angular distributions of electrons from the decay of 60 Co:
60
27 Co
ÿ
! 60
28 Ni ‡ e ‡ e :
The 60Co nucleus, of spin 5 », has a large magnetic moment, and the
nuclei in the sample were polarised by a magnetic ®eld. The electrons
from the decays were observed to be preferentially emitted in the opposite
direction to the nuclear spins. Such a correlation violates the principle of
mirror symmetry, since in the mirror image of this experiment, more
electrons appear to be emitted in the same direction as the nuclear spin.
An examination of Fig. 12.2 will make this clear. (Any mirror plane may
Fig. 12.2 A schematic representation of the 60Co decay experiment in real space
(a), and the mirror image of the experiment (b). In both (a) and (b) the spin of the
cobalt nucleus is pointing to the right; the spin is a pseudo-vector which does not
change direction under this re¯ection. The sample is polarised by the magnetic
®eld produced by a current ¯owing in the direction indicated.
166
Beta decay and gamma decay
be chosen, and will lead to the same conclusion.) Since the description of
the experiment and of its mirror image differ, it follows that parity cannot
be a symmetry of the weak interaction (}2.6). The breakdown of parity
conservation in the decay of the muon, described in }2.6, was discovered
shortly afterwards.
12.2
The Fermi theory of þ-decay
A simple theory of þ-decay was suggested by Fermi in 1934. Although
this theory is incomplete (it does not allow for parity violation, for example), it is able to describe the spectra of Fig. 12.1, and gives a qualitative
understanding of the range of þ-decay mean lives.
To be speci®c, we consider in the shell model a decay in which a
proton outside a doubly closed shell changes to a neutron:
17
9F
! 178 O ‡ e‡ ‡ e :
A proton in the closed shell cannot change into a neutron since the
neutron shell is also full and the Pauli principle forbids the transition.
Thus the nucleons in the closed shells play no part in the decay and we
can take the initial state of the system to be simply
þ0 ˆ ýp rp †;
where ýp is the state of the single proton in the d52 shell. The ®nal state of
the system consists of a neutron in the same shell, a positron e‡ , and a
neutrino e ,
þf ˆ ýn rn †ýe re †ý r †;
(in an obvious notation). Note that we are ignoring the spins of the
particles involved and neglecting the recoil of the nucleus.
The transition rate from þ0 to þf is given in perturbation theory by:
transition rate =
2
jHf0 j2 nf E0 †;
»
12:1†
where Hf0 is the matrix element linking the initial and ®nal states, and
nf E0 † is the density of (speci®ed) states þf at the energy E0 released in the
decay. This result is obtained in Appendix D (equation (D.6)) and is often
called `Fermi's golden rule' in texts on quantum mechanics.
12.2 The Fermi theory of þ-decay
167
We saw in Chapter 2 that the weak interaction responsible for þdecay, mediated by the heavy bosons W , is of very short range
2 10ÿ3 fm. This fact was anticipated by Fermi, who suggested that
at the moment of interaction all particles were at the same point in space,
so that the interaction matrix element
Z
Hf0 ˆ
þf Hþ0 d3 rn d3 rp d3 re d3 rv
could be of the form
Z
Hf0 ˆ Gw
ýn r†ýe r†ý r†ýp r†d3 r;
where the constant Gw is a measure of the strength of the weak interaction.
We may take for the neutrino a plane wave state
ý r† ˆ
1
1
V2
eik r ;
with the wave-function normalised in an arbitrarily large volume V for
mathematical convenience. We take the positron wave-function to be also
a plane wave
ýe r† ˆ
1
1
V2
eike re ;
though this is only a ®rst approximation; since a positron (or electron) is
charged, its wave-function will be modi®ed by the Coulomb ®eld of the
daughter nucleus. The matrix element Hf0 becomes in this approximation
Hf0 ˆ
Gw
V
Z
ýn r†ýp r†eÿi ke ‡k †r d3 r:
The energies involved in þ-decay are generally at most a few MeV,
and the corresponding momenta »ke , »k a few MeV/c. Hence the wavevectors ke , k are MeV/ »c 10ÿ2 fmÿ1 , and the exponent of the exponential in the integral is small over the range of the nuclear wave-functions. It is therefore an excellent approximation to expand the
exponential to give
168
Beta decay and gamma decay
Z
Gw
ýn r†ýp r†d3 r
Hf0 ˆ
V
Z
iG
ÿ w ke ‡ k † ýn r†rýp r†d3 r ‡ ;
V
and keep only the ®rst non-vanishing term. This argument is clearly also
valid when the nuclear wave-functions involved are more complicated
than in our simple example.
A decay is said to be allowed if the ®rst term is ®nite. It is said to be
®rst forbidden if the ®rst term is zero as happens for example if the initial
and ®nal nuclear states are of opposite parity, but not the second, and so
on. The diversity of þ-decay rates is largely accounted for by the degree of
forbiddenness of the transition and this in turn by the change in nuclear
spin (as in ÿ-decay). In the case of indium, previously cited, the ®rst term
in the expansion not to vanish is found to be the ®fth and the decay is
fourfold forbidden.
We shall concentrate our discussion on allowed transitions, in which
case the matrix element is
Hf0 ˆ Gw =V†MF ;
12:2†
where MF is the appropriate nuclear matrix element. In our example of
the decay of 19F, the spatial shell model wave-functions of the proton and
neutron are the same apart from Coulomb effects, and hence in this
simpli®ed theory
Z
MF ˆ
12.3
ýn r†ýp r†d3 r 1:
Electron and positron energy spectra
Consider an allowed þ-decay in which the electron is emitted in a particular state of (relativistic) energy Ee . For simplicity we neglect the recoil
energy of the daughter nucleus, which is in any case always a small
correction. Then the relativistic neutrino energy E is given by
E0 ˆ Ee ‡ E , where E0 =c2 is the nuclear mass difference. The density
of ®nal states factor in the formula (12.1) is thus the density of neutrino
states, n E0 ÿ Ee †, at energy E0 ÿ Ee †. There are ne Ee †dEe electron
states with energies between Ee , Ee ‡ dEe , where ne is the density of
12.3 Electron and positron energy spectra
169
electron states. Thus the total transition rate dR for decays to electron
states with energies in the range Ee , Ee ‡ dEe , is
dR ˆ
2
jHf0 j2 n E0 ÿ Ee †ne Ee †dEe :
»
12:3†
We need expressions for the densities of states n , ne . As is explained in
Appendix B, neglecting spin there are ‰V= 2†3 Š4k2 dk plane wave states
with jkj in the range k, k ‡ dk. For the relation between E and k we must
use for both electrons and neutrinos the relativistic formula
E 2 ˆ p2 c2 ‡ m2 c4 ˆ »k†2 c2 ‡ m2 c4 ;
1
so that E dE ˆ »2 c2 k dk, k ˆ E 2 ÿ m2 c4 †2 = »c. Thus the relativistic density of states formula for a particle of mass m is
n E†dE ˆ
1
V 4
E 2 ÿ m2 c4 †2 E dE:
3 3 3
2† » c
12:4†
Note that E is here the total energy, which includes the rest-mass energy.
Substituting in equation (12.3) and using (12.2) for the matrix element of
an allowed transition gives
dR ˆ
G2w jMF j2
S0 Ee †dEe
23 »7 c6
12:5†
where
1
1
S0 Ee † ˆ ‰ E0 ÿ Ee †2 ÿ m2 c4 Š2 E0 ÿ Ee † Ee2 ÿ m2e c4 †2 Ee :
The arbitrary normalisation volume V cancels out from the ®nal result
(12.5), as we should expect. We have allowed the neutrino mass to be
®nite in order to discuss the direct experimental evidence for its small or
zero value.
The electron (positron) energy dependence in the transition rate
(12.5) comes entirely from the lepton density-of-state factors included
in S0 E†: the other factors are independent of electron energy. The formula can be improved by allowing for the interaction between the electron and the Coulomb ®eld of the daughter nucleus of charge Zd . Since
only the electron wave-function at the nucleus is important, S0 Ee † is
modi®ed to
170
Beta decay and gamma decay
Sc Ee † ˆ F Zd ; Ee †S0 Ee †;
12:6†
where
þ
þ
þýe Zd ; 0†þ2
þ ;
þ
F Zd ; Ee † ˆ þ
ýe 0; 0† þ
and ýe Zd ; r† is the electron wave-function at energy Ee in the Coulomb
potential Zd e2 =4"0 r. Extensive tables of F Z; e† are available for precise calculations, but a simple approximation is the non-relativistic formula
F Z; Ee † ˆ
2
;
1 ÿ eÿ2
where ˆ Ze2 = 4"0 »v†; the positive sign holds for electrons, the negative for positrons, and v is the electron (positron) ®nal velocity.
As v ! 0, F Z; Ee † ! 2 for electrons. The 1=v† factor makes
Sc Ee † non-vanishing at the origin, where Ee ! me c2 ‡ 12 me v2 ; the
decay rate is enhanced at low energies since the Coulomb ®eld for electrons is attractive.
For positrons at low energies F Z; Ee † ! 2jjeÿ2jj . The Coulomb
®eld is repulsive for positrons and we can recognise the exponential as the
tunnelling factor through the Coulomb barrier, which suppresses positron
emission at low energies.
Figure 12.1 shows ®ts to the experimental statistical spectra for electron and positron emission from 64Cu. The Coulomb-corrected Sc Ee †
give excellent agreement with the shapes of the experimental curves.
The more detailed theory of þ-theory retains this factor.
The total transition rate for a particular allowed decay is obtained by
integrating the partial rate (12.5) over all electron energies, to give for the
mean lifetime the formula
1 G2w jMF j2 m5e c4
f Zd ; E0 †;
ˆ
23 »7
where
12.4 Electron capture
f Z; E0 † ˆ
1
me c2
5 Z
E0
mc2
171
1
F Z; Ee † E0 ÿ Ee †2 Ee2 ÿ m2e c4 †2 Ee dEe :
12:7†
To obtain (12.7) we have set m ˆ 0. f Z; E† is a dimensionless function
for which again there are extensive tables. Some representative graphs are
given in Fig. 12.3.
12.4
Electron capture
In an atomic environment, þ-decay by electron capture always competes
with positron emission, and is sometimes the only energetically allowed þdecay. To take our previous example, 179 F can also decay by electron
capture:
17
9F
‡ eÿ ! 178 O ‡ e :
The electron and proton wave-functions now constitute the initial state,
Fig. 12.3 The function f Z; E0 †. The sequence of curves is for Z ˆ 90, 60, 30 and
0 for eÿ decay and continuing with Z ˆ 30, 60, 90 for e‡ decay. (Formulae can be
found in Feenberg, E. & Trigg, G. (1950), Rev. Mod. Phys. 22, 399.)
172
Beta decay and gamma decay
þ0 ˆ ýp rp †ýe re †;
and the electron is most likely to be a K-shell electron, since K-shell wavefunctions have the greatest overlap with the nucleus. To a good approximation, these wave-functions are hydrogen-like, little in¯uenced by the
outer shell atomic electrons, so we can take
ÿ
ýe r† ˆ ÿ12
Zme e2
4"0 »2
!32
ÿ
!
Zme e2 r
exp ÿ
;
4"0 »2
where Z is the atomic number of the parent nucleus. The ®nal state is
þf ˆ ýn rn †ý r †:
1
For the neutrino we again take a plane wave state V ÿ2 eikr normalised in a
volume V.
In the simple Fermi theory, we now have
Z
Hf0 ˆ Gw
ýn r†ý r†ýp r†ýe r†d3 r:
Assuming that the transition is allowed, this reduces to
Hf0 ˆ
Gw
1
V2
Z
ýe 0†
ýn r†ýp r†d3 r ˆ
Gw
1
ÿ
1
V 2 2
Zme e2
4"0 »2
!32
MF ;
since the electron and neutrino wave-functions can be treated as constant
over the nuclear volume.
Neglecting the nuclear recoil, the emitted neutrino has energy E ,
where E =c2 is the atomic mass difference E0 =c2 ‡ me † (cf. (4.13)).
The appropriate density of states in the formula for the transition rate
is the neutrino density of states at this energy, given by equation (12.4).
Setting m ˆ 0,
n E † ˆ
and we obtain
V 4 2
E ;
2†3 »3 c3
12.5 The Fermi and Gamow±Teller interactions
173
2
jHf0 j2 n E †
»
ÿ
!3
G2w jMF j2 E2 Zme e2
ˆ
:
2 »4 c3
4"0 »2
decay rate for electron capture ˆ
Consistently with our neglect of electron spin, only one K electron is
included in the calculation. The ratio of the electron capture rate RK to
the positron emission rate Re‡ is independent of Gw and the nuclear
matrix element, and is
RK
E 2 Z 3
1
ˆ 2
:
2
137
f
Z
Re‡
me c
d ; E0 †
1
. For low values of Z this
Note that E E0 ‡ me c2 , and e2 =4"0 »c 137
3
ratio is usually small, but at high Z the Z factor, and the increasing
Coulomb barrier for positron emission which reduces f Zd ; E0 †, make
electron capture the dominant process.
12.5
The Fermi and Gamow±Teller interactions
In the simple Fermi theory of þ-decay, the interaction matrix element was
written as a `contact' interaction. For our example of 17F decay,
Z
þf Hþi d coordinates†
Z
ˆ Gw ýn r†ýe r†ý r†ýp r†d3 r;
Hf0 ˆ
which we might represent diagrammatically as in Fig. 12.4. Reference to
spin has been suppressed, though we know that the particles involved are
all fermions with intrinsic spin quantum number s ˆ 12.
In the full theory of þ-decay, the interaction is mediated by the
charged W bosons, so that the process above is represented by Fig.
12.5. At a more fundamental level, the interaction is with a quark rather
than a nucleon, as in Fig. 3.3, but phenomenologically the principal
missing feature of the simple Fermi theory is the description of spin
effects. We shall now describe how the results of the previous section
are modi®ed when spin is taken into account. The nucleon states in our
example can still be described non-relativistically, but in general include
`spin-up' and `spin-down' contributions (see Appendix C, }C.2):
174
Beta decay and gamma decay
Fig. 12.4 þ-decay of a proton in a nucleus as a `contact' interaction.
1
0
ý‡
;
ý r† ˆ ý‡ r†
‡ ýÿ r†
ˆ
ýÿ
0
1
and hence are two-component wave-functions. The complex conjugate
wave-function generalises to the adjoint row matrix ýy r† ˆ ý‡ ; ýÿ †.
However, both the positron and neutrino move at relativistic speeds
and for these the relativistic wave-functions must be used. Except in
terms of the Dirac wave-functions, there is no simple form for the lepton
part of the matrix element.
The contribution to the interaction from the Coulomb-like part of the
W-®eld is most like the simple Fermi theory discussed in the previous
sections and it is called the Fermi interaction. This part does not change
the nucleon spins, and for allowed transitions the positron and neutrino
angular momenta must combine to give a total lepton angular momentum of zero.
The contribution of the Fermi interaction to the interaction matrix
element is
F
Hf0
Z
ˆ Gw
y
ýn r†ýp r†d3 r lepton part†:
The subtlety of the weak interaction is contained in the bracketed
lepton part. This involves the neutrino and positron wave-functions eval-
Fig. 12.5 þ-decay of a proton in a nucleus mediated by the exchange of a virtual
W boson.
12.5 The Fermi and Gamow±Teller interactions
175
uated at the nucleon coordinate r, as in the simple Fermi theory, but also
describes the alignment of the neutrino and positron spins and the angular correlation between their directions. (The neutrino direction can be
inferred by measuring the small nuclear recoil.) The lepton part is given
very precisely by the Standard Model of particle physics and books on
this subject should be consulted for detailed calculations. However, an
experiment which only measures the electron energy spectrum and does
not distinguish these correlations corresponds to an averaging over directions and spins, and then the spectrum is given exactly as in the simple
theory. If only the Fermi interaction contributes to the decay, the energy
spectra, decay rates, mean lives and electron capture rates are given by the
previous formulae, but with the nuclear matrix element given by
Z
MF ˆ
ýyn r†ýp r†d3 r:
The constant Gw is given in terms of more fundamental constants of
particle physics by
Gw ˆ GF Vud :
Here GF is the Fermi constant and Vud is an element of the `Kobayashi±
Maskawa matrix', which appears in the Standard Model theory of the
weak interaction between leptons and quarks. GF may be determined
experimentally from the decay rate of the muon and has the value
GF ˆ 1:166 39 2† 10ÿ11 »c†3 MeVÿ2 :
12:8†
From a range of nuclear data, it is found that
Vud ˆ 0:9744 10†:
But this is not the whole story, even for allowed transitions. The
magnetic-like part of the W-®eld leads to a term in the transition matrix
element, known as the Gamow±Teller interaction, in which the total
lepton angular momentum J has quantum number j ˆ 1, and the nuclear
part of the interaction (again treated non-relativistically) contains the
Pauli operator r (see Appendix C). There is a term r J in the interaction
Hamiltonian. The Gamow±Teller matrix element for our 17F example is
176
Beta decay and gamma decay
GT
Hf0
Z
ˆ gA Gw
y
ýn r†rýp r†d3 r lepton part†;
where (lepton part) is now a vector and denotes a scalar product.
If we de®ne
Z
MGT ˆ gA
y
y
x
z
; MGT
; MGT
†;
ýn r†rýp r†d3 r ˆ MGT
and sum over all allowed decays to the j ˆ 0 and the three j ˆ 1 states, the
total decay rate to electrons with energies in the range Ee , Ee ‡ dEe is
given by
dR Ee † ˆ
2
G2F Vud
y 2
x 2
z 2
‰jMF j2 ‡ jMGT
j ‡ jMGT
j ‡ jMGT
j ŠSc Ee †dEe ;
23 »7 c6
and the mean life is given by
2
1 G2F Vud
m5e c4
y 2
x 2
z 2
‰jMF j2 ‡ jMGT
j ‡ jMGT
j ‡ jMGT
j Šf Zd ; E0 †:
ˆ
23 »7
12:9†
An allowed decay may be pure Fermi, pure Gamow±Teller, or a
mixture of both, depending on the details of the nuclear matrix elements.
Note that the electron energy spectrum is independent of these details. In
general, of course, the initial and ®nal nuclear states which enter into MF
and MGT are more complicated than those of our 17F example. MF and
MGT always vanish if the initial and ®nal nuclear states are of opposite
parity, since r is an axial vector. Thus there can be no parity change in the
nuclear states in an allowed transition.
The axial coupling constant gA which appears in the expressions above
is a parameter of the theory. It is not a fundamental particle physics
parameter. We shall see in }12.6 that gA 1:26.
For a Fermi transition, the change j in nuclear spin must be zero.
For a Gamow±Teller transition, j ˆ 0 or 1, by the rules for addition of
angular momentum, except that 0 ! 0 transitions are forbidden since the
matrix element of r vanishes between two spherically symmetrical states.
12.6 The constants Vud and gA
12.6
177
The constants Vud and gA
In the decay
14
8O
!147 N ‡ e‡ ‡ e
the transition occurs, with 99.7% probability, to the ®rst excited state of
the daughter nucleus, which has spin and parity 0‡ . The even±even
nucleus 148 O also has spin and parity 0‡ , so that from the selection rules
above the transition is allowed, and pure Fermi. Also, in the nuclear shell
model, the nuclei differ only in that 14O has two protons in 1p12 states
outside a 126 C core, and 14N* has one proton and one neutron. Thus,
because of the charge independence of the strong nuclear force and the
smallness of the Coulomb effects in these light nuclei, the wave-functions
of the initial and ®nal nuclear states are very similar, and jMF j2 2 (since
either of the two protons in 1p12 states can decay). The energy
E0 ˆ 2:32 MeV, Zd ˆ 7 and f 7; 2:32† ˆ 42:8. The measured mean life
is 102 s. Thus from the formula (12.9) for the mean life we can calculate
GF Vud = »c†3 ˆ 1:16 10ÿ11 MeVÿ2 :
From (12.8) we see immediately that Vud is close to unity. This mean life
measurement gave one of the ®rst estimates of the particle physics parameter Vud .
The constant gA is most directly determined from the lifetime of a free
neutron, since there are then no uncertainities in the computation of
nuclear wave-functions. Indeed, if we neglect recoil, the spatial parts of
the initial neutron and ®nal proton wave-functions are the same. Suppose
the spin state of the neutron is j ‡ 12i. If the proton spin state is j ‡ 12i then,
using the properties of the r matrices (Appendix C), MF ˆ 1 and
MGT ˆ gA Gw † 0; 0; 1†. If the proton spin state is j ÿ 12i, MF ˆ 0 and
MGT ˆ gA Gw † 1; i; 0†. The neutron±proton mass difference gives
E0 ˆ 1:29 MeV and f 1; 1:29† ˆ 1:6. The total decay rate to all possible
spin states is therefore, from equations (12.6) and (12.8), given by
2
1 1:6‰1 ‡ 3g2A ŠG2F Vud
m5 c4
:
ˆ
23 »7
The measured mean life of the neutron is 887 s, which yields
gA ˆ 1:3, using this formula. However, there are `radiative corrections'
178
Beta decay and gamma decay
to our simple expression and the present accepted value of gA is a little
lower,
gA ˆ 1:26:
12.7
Electron polarisation
The lepton part of the interaction matrix element leads to angular correlations between the various spins and momenta of the four particles
involved in a þ-decay. These correlations can be detected in suitable
experiments, as for example the spin-polarised 60Co experiment discussed
in }12.1; the observed angular distribution of electrons in this experiment
is in accord with the theory.
The non-parity conserving nature of the weak interaction is most
clearly seen in the lepton states. All neutrinos are `left-handed' and all
anti-neutrinos `right-handed'. The theory also predicts that in þ-decay
left-handed electrons are produced more copiously than right-handed
electrons, whereas positrons produced in þ-decay are predominantly
right-handed. More precisely, the probability of an electron emitted
with velocity v being in a left-handed state (with intrinsic spin s antiparallel to momentum p) is
PL ˆ
1
v
1‡ ;
2
c
and the probability of its being emitted in a right-handed state (with s
parallel to p) is
PR ˆ
1
v
1ÿ :
2
c
Hence
Pˆ
PR ÿ PL
v
ˆÿ
PR ‡ PL
c
(and for positrons P ˆ ‡v=c†.
Figure 12.6 shows experimental measurements of P, the `longitudinal
polarisation', plotted against v=c for a variety of þ-decays. The complete
polarisation of the neutrino and anti-neutrino can be regarded as a generalisation of this result, since v ˆ c for massless particles.
12.8 Theory of ÿ-decay
179
Fig. 12.6 Measured degree of longitudinal polarisation P for allowed eÿ decays.
(Data from Koks, F. W. J. & van Klinken, J. (1976), Nucl. Phys. A272, 61.)
12.8
Theory of ÿ-decay
In ÿ-decay, a nucleus in an excited state falls to a lower state with the
emission of a photon (}7.3). The electromagnetic interaction which governs this process is very well understood theoretically, but a full discussion requires the quantised equations of the electromagnetic ®eld, rather
than the classical Maxwell equations, and is beyond the scope of this
book. However, we can understand the main features of ÿ-decay, and
in particular the great range of ÿ-decay lifetimes described in }7.3, using
semi-classical arguments to write down an approximate expression for the
interaction energy between a nucleus and a photon.
We again enclose our system in a large volume V. Consider the plane
electromagnetic wave
E ˆ E0 cos k r ÿ !t†;
B ˆ B0 cos k r ÿ !t†:
The standard Maxwell theory tells us that in such a wave jBj ˆ jEj=c, and
the energy is divided equally between the electric and magnetic ®elds and
is given in total by
180
Beta decay and gamma decay
Z
"0
E r† E r†d3 r ˆ 12"0 E20 V;
since the cosine squared averages to 12 over the volume V. If we identify
this wave with a single photon of wave-vector k and energy »! we must
therefore set
2
1
2"0 E0 V
ˆ »!;
1
or
jE0 j ˆ 2 »!="0 V†2 :
In a typical ÿ-decay, »! is at most a few MeV, so that
jkj ˆ »!= »c 1 MeV†= 197 MeV fm† 10ÿ2 fmÿ1 . Hence to a good
approximation we can neglect the change in k r† over the dimensions
of the nucleus (fm), which we can take to be centred at r ˆ 0. The
electric ®eld over the nucleus is then
E ˆ E0 cos !t ˆ 12E0 ei!t ‡ eÿi!t †;
and the potential energy of the nucleus in such a ®eld is given classically
by
X
ÿe
protons
E rp ˆ ÿ
e X
E r ei!t ‡ eÿi!t †:
2 protons 0 p
In a ÿ-decay, we start with a state in which there is no photon present,
and end with a state in which there is one photon present and the nucleus
is in a lower energy state. As in our discussion of þ-decay, we shall neglect
the small nuclear recoil energy. It is clear from the derivation of the result
(D.6) in Appendix D that only the term with ei!t in the interaction can
contribute to this transition, so that the matrix element to be employed in
the formula for the decay rate is
e
ÿ
2
Z
ÿ
þf
X
!
e
E0 rp þ0 d coordinates† ˆ ÿ E0 Rf0 ;
2
protons
where þ0 , þf are the initial and ®nal nuclear states and
Z
Rf0 ˆ
ÿ
þf
X
protons
!
rp þ0 d coordinates†:
12:10†
If Rf0 is non-vanishing the transition is said to be electric dipole (E1).
12.8 Theory of ÿ-decay
181
Let us assume Rf0 is non-vanishing and real. The treatment is easily
extended to the case when Rf0 is a complex vector R1 ‡ iR2 since
jE0 R1 ‡ iR2 †j2 ˆ jE0 R1 j2 ‡ jE0 R2 j2 :
For a given direction of photon emission, there are two independent
photon states with polarisations which we can take as in the plane de®ned
by Rf0 and k, and perpendicular to this plane (Fig. 12.7). For the latter
E0 Rf0 ˆ 0, so that the transition probability to this state vanishes.
If is the angle between the direction k and Rf0 , for the state with
polarisation in the plane we have
jE0 Rf0 j ˆ jE0 jjRf0 j sin since E0 is perpendicular to k. The density of states at energy Eÿ ˆ »!,
for photons emitted in a solid angle dÿ ˆ sin d d, is
V 2 dk
V !2
k
dÿ
ˆ
sin d d;
2†3 dEÿ
2†3 »c3
since Eÿ ˆ »! ˆ »ck. Hence the `Fermi golden rule' formula gives the
transition rate
2 e2
V !2
jE0 j2 jRf0 j2
sin2 dÿ:
» 4
2†3 »c3
12:11†
There is a characteristic sin2 angular distribution of the emitted
photons. Such angular distributions can be observed experimentally, if
for example the nuclei in a sample are oriented in the same direction by a
strong magnetic ®eld.
Fig. 12.7 Direction of emission k and polarisation vector E0 for an allowed
electric dipole transition.
182
Beta decay and gamma decay
The mean life is obtained by integrating the expression (12.11) over
1
all directions in space. Using jE0 j ˆ 2 »!="0 V†2 we obtain
ÿ
!
1
4 e2
!3
ˆ
jR j2
E1 3 4"0 »c3 f0
3 Eÿ
jRf0 j 2
ˆ
1 MeV
1 fm
12:12†
0:38 10
15
ÿ1
s ;
where the last form indicates the order of magnitude to be expected for
the mean lives.
From equation (12.10), we see that þ0 and þf must be of opposite
parity for an electric dipole transition to take place, since if they have the
same parity the integral vanishes. It can also be shown from the angular
part of the integration, using the properties of spherical harmonics, that
the change j in the nuclear spin quantum number for an electric dipole
transition must be j ˆ 0 or j ˆ 1, except that 0 ! 0 transitions are
forbidden (see }7.3). An estimate of the magnitude of jRf0 j requires a
knowledge of the nuclear wave-functions. Even in the simple shell
model such calculations are not easy.
The nucleus also couples to the magnetic ®eld of the photon, and at a
similar level of approximation the interaction with the magnetic ®eld B is
given classically by ÿl B0 cos !t, where l is the total magnetic moment
of the nucleus. The magnetic moment operator is given in the simple shell
model by equation (5.24),
lˆ
X
N ‰gL L ‡ gs sŠ= »;
nucleons
where N ˆ e »=2mp is the nuclear magneton. The transition rate induced
by this interaction will be of the same form as (2.11), with eE0 Rf0
replaced by B0 Mf0 where
Z
Mf0 ˆ
þf lþ0 d coordinates†:
12:13†
If Mf0 is non-vanishing, the transition is said to be magnetic dipole (M1).
Since jB0 j ˆ jE0 j=c, the mean life is given by
1
M1
3
4
1
!
ˆ
jM j2 :
3 4"0 »2 c5 f0
12:14†
12.8 Theory of ÿ-decay
183
From (12.13), Mf0 is non-vanishing only if þ0 and þf have the same
parity, since l is a pseudo-vector (}2.6). Electric and magnetic dipoles
transitions are therefore mutually exclusive. The angular momentum
selection rules, j ˆ 0, 1 0 ! 0 forbidden) are the same as for electric
dipole transitions.
The ratio of mean lives for magnetic and electric dipole transitions at
the same energy is
M1 e2 c2 jRj2
ˆ
:
E1
jMj2
1
If we take jRj nuclear radius A3 fm, and M e »=mp , we obtain
2
mp c2 †2 A3 fm2 †
2
M1
20A3 :
2
E1
»c†
Thus the mean lives for magnetic dipole transitions are generally longer
than those of electric dipole transitions at the same energy by a considerable factor, though this estimate is of course very crude.
If both Rf0 and Mf0 vanish, as is not uncommon, then we can no
longer neglect the variation in the photon ®eld over the dimensions of
the nucleus. The expansion of cos k r ÿ !t† in powers of k r† gives
matrix elements for higher-order electric and magnetic transitions.
Each power of k r† introduces an additional factor of ÿ1 in the
parity selection rule, and an additional unit of orbital angular momentum in the j selection rule so that, for example, for electric quadrupole transitions there is no change in parity and j ˆ 2, 1, 0,
except that 0 ! 0 and 12 ! 12 transitions are both forbidden, by conservation of angular momentum. To each type of transition there corresponds a characteristic angular dependence and polarisation of
emitted ÿ-rays.
Each power of k r† reduces the order of magnitude of the matrix
element by a factor kR†, where R is the nuclear radius, and hence
increases the lifetime by a factor of kR†ÿ2 . For a 1 MeV photon and
A 50, kR†ÿ2 ˆ 0:24 104 . The curves of Fig. 7.6 have been drawn
using only a more sophisticated version of this argument, but they are
nevertheless a useful guide to the interpretation of experimental
lifetimes.
184
Beta decay and gamma decay
12.9
Internal conversion
A nucleus in an excited state can also decay electromagnetically by `internal conversion'. In this process, an atomic electron in a state 0 re † takes
up the energy released in the decay and is excited to a state f re † which
must be initially empty. If the energy release is greater than the binding
energy of the electron, as is usually the case, the electron is ejected from
the atom and the state f re † may be approximated by a plane wave.
Thus the initial state is of the form
þ0 ˆ ýnuc
0 0 r e †
and the ®nal state of the form
þf ˆ ýnuc
f f re †:
The main contribution to the interaction energy between the electron and
the nucleus is the Coulomb energy
X
ÿe2
4"0 jrp ÿ re j
protons
and the corresponding matrix element for the transition is
Z
Hf0 ˆ
ýnuc
f
f
X
ÿe2
4"0 jrp ÿ re j
protons
3
ýnuc
0 0 d re d nuclear coordinates†:
We shall not pursue the evaluation of this matrix element, but note that it
can be non-vanishing for 0 ! 0 transitions between nuclear states of the
same parity.
The process of internal conversion always competes with ÿ-decay
with similar nuclear matrix elements appearing. As in the case of Kcapture in þ-decay, there is a factor Z 3 in the transition rate arising
from the normalisation of the initial state electron wave-function. Thus
internal conversion becomes increasingly signi®cant in the electromagnetic decays of the heavier elements. The internal conversion coef®cient
is de®ned as the ratio of the rate of internal conversion to the rate of ÿemission, for a given type of electromagnetic transition. Extensive tables
of these coef®cients can be found in the literature.
Problems
185
Problems
12.1 The product of a þ-decay half life T12 (}2.4) and the number f Zd ; E0 † is
the `fT12 ' value. From equations (12.7) and (12.9) the fT12 value gives a
direct empirical determination of the nuclear matrix element
jMF j2 ‡ jMGT j2 . Calculate the fT12 value for the decay
31
31
‡
for which T12 ˆ 2:60 s, E0 ˆ 4:94 MeV and
16 S !15 P ‡ e ‡ ,
f 15; 4:94† ˆ 1830. In the simple shell model, this decay involves a 2s12
proton changing to a 2s12 neutron. Compare this fT12 value with that of a
free neutron. Why do the two values differ?
12.2 The cross-section for the reaction
e ‡ p ! n ‡ e‡
is given in perturbation theory by
ˆ
2
1
jH j2 n Ee †;
» neutrino flux† f0
where n Ee † is the relativistic density of states (equation (12.4)). Show
that
G2w
‰1 ‡ 3g2A Š cpe †Ee :
»c†4
Calculate this cross-section for a 2 MeV anti-neutrino.
12.3 If an electric dipole (E1) decay mean life is known, then equation (12.12)
can be used to calculate the corresponding dipole matrix element jRf0 j.
The ®rst excited state of 11Be decays to the ground state through an E1
transition. The mean life is 1:79 10ÿ13 s and the photon energy is
0.32 MeV. Calculate jRf0 j.
An example of an electric dipole transition in atomic physics is the
decay of the 2p excited state of the hydrogen atom, for which the mean
life is 1:6 10ÿ9 s and the photon energy is 10.2 eV. Calculate jRf0 j and
compare it with the nuclear matrix element.
‡
12.4 The nucleus 108
47 Ag, which has spin and parity 1 , is þ-unstable with a
mean life of 3.4 minutes. It has an excited state at 109 keV excitation
energy, spin and parity 6‡ , which is an isomeric state with a mean life of
180 years. Explain how an excited state of a nucleus can be more stable
than the ground state.
13
Neutrinos
Neutrinos are elusive particles: for many years their very existence was
only inferred from the part that they play in ÿ-decay. However, we have
seen in Chapter 10 and Chapter 11 that they are of great importance in
astrophysics, and in the forging of the nuclei of the heavy elements in
supernovae. Apart from ÿ-decay, other experimental results on neutrinos
are accumulating. In this chapter we describe some of these results, and
their possible interpretation.
13.1
Neutrino cross-sections
To design neutrino detectors, for example to measure the ¯ux of neutrinos striking the Earth, it is important to know their interaction crosssections with atomic nuclei and electrons. Unless the neutrino energy is so
high that its de Broglie wavelength ˆ 2 »=p† is comparable with or
less than the nuclear radius, the nuclear cross-sections for processes which
convert a neutrino to its charged lepton partner will involve matrix elements of the same form as those which appear in the theory of ÿ-decay.
For example the total cross-section for the reaction
e ‡ p ! n ‡ e‡ ;
with unpolarised protons, at a neutrino energy above the threshold
energy for the reaction, is (Problem 12.4)
ˆ
186
G2w
‰1 ‡ 3g2A Š cpe †Ee :
»c†4
13:1†
13.1 Neutrino cross-sections
187
In this expression pe and Ee are the electron momentum and electron
energy in the centre-of-mass system. (Note that the energy of the electron
is determined by the energy of the neutrino.)
More generally, the total cross-section for the reaction
e ‡ A ! B ‡ eÿ ;
when the nuclear matrix elements correspond to an allowed transition,
and the neutrino energy is above threshold, is given by
ˆ
þ2 þ x þ2 þ y þ2 þ z þ2 i
G2w hþþ
þ þ
þ þ
þ þ
þ cpe †Ee F Z; Ee †;
M
F ‡ MGT ‡ MGT ‡ MGT
»c†4
13:2†
where Z is the atomic number of the ®nal nuclear state, and F Z; E† is the
Coulomb correction factor introduced in equation (12.6).
The same formula holds for the anti-neutrino reaction,
e ‡ B ! A ‡ e‡ :
The reactions described above involve the exchange of a virtual W
boson. By the exchange of a virtual neutral Z boson, a neutrino can
scatter elastically from a nucleus. However for neutrino detection the
important elastic scattering is that from electrons (see }10.5). The processes involved in electron neutrino scattering from electrons are represented diagrammatically in Fig. 13.1. At low energies the W boson
contribution to the scattering comes from matrix elements of the form
Z
Hf0 ˆ GF
ef r†eo r†f r†o r†d3 r:
13:3†
Reference to spin has been suppressed. The form echoes the Fermi interaction in ÿ-decay, but since only leptons are involved it is the Fermi
constant GF which appears. The contribution from the interaction with
the Z boson involves a new parameter of the Standard Model: the
Weinberg angle W . For neutrino energies E me c2 (but less than
MW ) in the centre-of-mass system, the total cross-section for scattering
from unpolarised electrons is calculated to be
188
Neutrinos
Fig. 13.1 Processes contributing to the elastic scattering of electron neutrinos by
electrons: (a) exchange of a virtual charged W boson; (b) exchange of a neutral Z
boson.
ˆ
G2F
‰3 ‡ 12 sin2 W ‡ 16 sin4 W Š cpe †Ee :
3 »c†4
13:4a†
The corresponding cross-section for anti-neutrinos is
ˆ
G2F
‰1 ‡ 4 sin2 W ‡ 16 sin4 W Š cpe †Ee :
3 »c†4
13:4b†
In these expressions pe and Ee are the electron momentum and energy in
the centre-of-mass system. The accepted value of sin2 W from experiment
is
sin2 W ˆ 0:231 24 24†:
13.2
The mass of the electron neutrino
In standard uni®ed theories the neutrino masses are assumed to be zero.
It is clearly important to test this assumption experimentally. A ®nite
neutrino mass of even a few eV/c2 would have signi®cant consequences
in, for example, cosmology. The signature of a ®nite neutrino mass would
in ÿ-decay appear in the shape of the electron energy spectrum near
maximum energy. From expression (12.5), this shape depends sensitively
on whether m ˆ 0 or m 6ˆ 0. The difference is clearer in a Kurie plot of
13.3 Neutrino mixing and neutrino oscillations
"
189
#12
dR=dEe
1
F Zd ; Ee †Ee Ee2 ÿ m2e c4 †2
against electron energy since from equation (12.5)
"
#12
dR=dEe
1
F Zd ; Ee †Ee Ee2 ÿ m2e c4 †2
1
1
ˆ constant† E0 ÿ Ee †2 ‰ E0 ÿ Ee †2 ÿ m2 c4 Š4 :
If m ˆ 0, this plot gives a straight line E0 ÿ Ee † passing through E0 ; if
m 6ˆ 0 the line is curved and the tangent at maximum energy is vertical.
A much-studied decay in this context is that of tritium
3
1H
! 32 He ‡ eÿ ‡ ‡ 18:6 keV:
The low electron kinetic energies in this decay are experimentally advantageous. Figure 13.2 shows experimental data and there is remarkable
overall agreement between the data and the ®tted theoretical spectrum.
A Kurie plot of data near Ee ˆ E0 is also shown. The dif®culty of the
experiment is evident: the conclusion is that m < 60 eV=c2 . More recent
tritium experiments (Belesev, A. I. et al. (1995), Phys. Lett. B 350, 263)
suggest that
m < 4:35 eV
with high probability.
We conclude there is not yet direct evidence that the electron neutrino
has mass, and there is no direct evidence of mass for the muon neutrino
or the tau neutrino. However, there is a growing body of experimental
results which suggest that neutrinos do have mass, but on a much smaller
scale than that probed by direct experiments.
In }2.4 and }3.6 we introduced all of the elementary fermions: the
leptons and the quarks. The three charged leptons and the six charged
quarks all have mass. Are the neutrinos the exception?
13.3
Neutrino mixing and neutrino oscillations
Let us assume there are three basic neutrino types, which we denote by
j1 i, j2 i, j3 i, having de®nite masses m1 , m2 , m3 respectively. These states
190
Neutrinos
Fig. 13.2 The electron energy spectrum from the decay of tritium. The experimental points give the number of electrons N Ee † observed in small energy `bins'
from a very large number of decays. (Taken from Lewis, V. E. (1970), Nucl. Phys.
A151, 120.) The spectrum is well ®tted using equations (12.5) and (12.6). Also
shown for comparison is the curve without the Coulomb correction. The inset
shows a Kurie plot of the spectrum near the electron end point. (For this data see
Bergkuist, K. E. (1972), Nucl. Phys. B39, 317.) The theoretical curves in the inset
include the effect of the ®nite size of the energy `bins'.
are eigenstates of the mass operator, and may be taken to be orthogonal
and normalised. The mass operator acts on internal degrees of freedom of
the neutrino, and the mass eigenstates may be represented by 3 1 column matrices. The formalism is somewhat similar to that of intrinsic
electron spin, in which `spin-up' and `spin-down' states are eigenstates
of the operator sz .
13.3 Neutrino mixing and neutrino oscillations
191
Mixing means that the electron neutrino state je i produced in a ÿdecay is not a basic neutrino, but a linear combination of the three mass
eigenstates. Similarly the muon neutrino and the tau neutrino are linear
combinations of the mass eigenstates, orthogonal to each other and to the
electron neutrino.
For mathematical simplicity we shall take a two-component model
and suppose that je i is well represented by
je i ˆ cos j1 i ‡ sin j2 i:
13:5†
Orthogonal to je i is the state
jx i ˆ ÿ sin j1 i ‡ cos j2 i;
13:6†
as the reader may readily verify. jx i might be a muon neutrino or a tau
neutrino. is known as the mixing angle.
If the electron neutrino was created at t ˆ 0 with momentum p in the
plane wave state eipz= » , then at time t the state will have evolved to
13:7†
je it ˆ eÿiE1 t= » cos j1 i ‡ eÿiE2 t= » sin j2 i;
q
q
where E1 ˆ p2 c2 ‡ m21 c4 , E2 ˆ p2 c2 ‡ m22 c4 .
We can solve equations (13.5) and (13.6) for j1 i and j2 i in terms of
je i and jx i:
j1 i ˆ cos je i ÿ sin jx i;
j2 i ˆ sin je i ‡ cos jx i;
and then substitute in (13.7) to obtain
je it ˆ eÿiE1 t ‰ cos2 ‡ ei E1 ÿE2 †t= » sin2 †je i
ÿ sin cos 1 ÿ ei E1 ÿE2 †t †jx iŠ:
If after time t the neutrino is detected, the probability of it being an
electron neutrino is, by the usual rules of quantum mechanics,
Pe t† ˆ j cos2 ‡ ei E1 ÿE2 †t= » sin2 †j2
ˆ 1 ÿ sin2 2 sin2 f E2 ÿ E1 †t=2 »g;
and the probability of it being an x neutrino is
13:8†
192
Neutrinos
Px t† ˆ sin2 cos2 j 1 ÿ ei E1 ÿE2 †t= » †j2
ˆ sin2 2 sin2 f E2 ÿ E1 †t=2 »g:
13:9†
Unless m1 ˆ m2 , so that E1 ˆ E2 , it is evident that Pe < 1 if there is
mixing.
A neutrino may be detected and identi®ed by its conversion into its
charged leptonic partner (}10.5). In the case of neutrinos produced from
nuclear ÿ-decay, or from thermonuclear reactions in the Sun, the neutrino
energies are too small to produce a muon or tau lepton. Thus the x
neutrino will not be detected. If there is neutrino mixing, it will be seen
as a reduction by a factor Pe in the anticipated ¯ux of electron neutrinos.
Note that electron lepton number is no longer conserved (see }2.5): we are
straying outside the Standard Model of particle physics.
To analyse Pe further, we assume that the neutrinos are highly relativistic. We can then take the ®rst term in the Taylor expansion and write
E2 ÿ E 1 ˆ
q q
m2 †c4
;
p2 c2 ‡ m22 c4 ÿ p2 c2 ‡ m21 c4 2pc
13:10†
where m2 ˆ m22 ÿ m21 .
A relativistic neutrino wave packet will travel with velocity c so
that, at a distance z from the source,
Pe z† ˆ Pe t ˆ z=c† ˆ 1 ÿ sin2 2 sin2 f E2 ÿ E1 †z=2 »cg:
13:11†
Using (13.10) this becomes
Pe z† ˆ 1 ÿ sin2 2 sin2 z=L†;
13:12†
where the oscillation length L is given by
Lˆ
4 pc† »c†
;
m2 †c4
13:13†
which can be written as
pc L ˆ 2:48
1 MeV
ý
!
1 eV2
m:
m2 †c4
13:14†
13.4 Solar neutrinos
193
Table 13.1. The ratio of measured neutrino ¯ux to neutrino ¯ux predicted
by the standard solar model, in ®ve independent experiments
Experiment
Type of detector
Result/Theory
Homestake (USA)
Kamiokande (Japan)
GALLEX (Italy)
SAGE (Russia)
SuperKamiokande (Japan)
37
0:33 0:029
0:54 0:07
0:60 0:06
0:52 0:06
0:474 0:020
Cl
H2O
71
Ga
71
Ga
H2O
(See Bahcall, J. N. et al. (1998), Phys, Rev. D 58, 096016.)
It is the sinusoidal nature of Pe as a function of z which gives rise to
the name neutrino oscillations for this phenomenon.
13.4
Solar neutrinos
Table 13.1 shows the results of ®ve independent measurements of the
ratio of the solar neutrino detection rate, to the expected rate calculated
from the standard solar model and the Standard Model of particle physics. Within the context of neutrino oscillations, this can be taken as a
measure of Pe . Pe is consistently less than one. Also, the measurements
are sensitive to different neutrino energy bands (}10.5), and Pe appears to
be energy dependent.
Can these results be explained by neutrino oscillations? Within our
two-neutrino model, if m2 †c4 1 eV2 we see from equation (13.14) that
the oscillation length L is then very much smaller than the size of the
Sun's thermonuclear core, where solar neutrinos are created. (For solar
neutrinos, pc 10 Mev.) Averaging over the core,
Pe ˆ 1 ÿ 12 sin2 2:
This is independent of neutrino energy, and the maximum possible suppression is Pe ˆ 12. Thus m2 †c4 1 eV2 would appear to be inconsistent
with the experimental data.
An alternative relevant length scale is the mean Earth±Sun distance of
1:496 1011 m ˆ 1 astronomical unit. We can rewrite (13.12) as
194
Neutrinos
"
Pe z† ˆ 1 ÿ sin2 2 sin2
ý
!#
z 1 MeV m2 †c4
1:9
1 AU
pc
10ÿ11 eV2
13:15†
and 0 < pc 10 MeV, z ˆ 1 AU (ignoring the diurnal and annual variations of z). Unless m2 †c4 10ÿ11 eV2 , the range of neutrino
momentum values will lead to an averaging of the sine-squared term,
and again we shall have a factor of 12. However, a detailed analysis
suggests that the experimental data can be ®tted by this model if
m2 †c4 6:5 10ÿ11 eV2 , and sin2 2 0:75.
Another interesting possible mechanism for explaining the solar neutrino de®cit arises from the interaction of neutrinos with matter. As
neutrinos pass through matter they acquire an effective mass. An electron
neutrino can interact with electrons in matter through the exchange of a
virtual W boson, as in Fig. 13.1(a). At low energies this gives the Fermi
interaction matrix elements (13.3). These can be interpreted as arising
from a neutrino effective mass GF ne =c2 , where ne is the numberpdensity

operator for electrons. Taking spin into account gives a factor 2. The
effective mass from the Fermi interaction is thus given by
me c2 ˆ
p
3
2ne GF ˆ 1:27 1 A ne † 10ÿ13 eV:
13:16†
The Gamow±Teller interaction contributes only in ferromagnetic materials. This effective mass term is unique to the electron neutrino, because
matter has zero muon density and zero tau density. The neutrino interaction with electrons through the neutral Z boson (Fig. 13.1(b)) contributes to the effective mass for all neutrino types, but does not induce mass
differences. It is therefore of little interest in the context of neutrino
oscillations.
These effective masses are calculated in the Standard Model of particle physics. They are small, both in terrestrial materials and in the Sun,
as equation (13.16) indicates. However, if we postulate the existence of
intrinsic neutrino masses and neutrino mixing, new resonance phenomena
appear. In passing through matter of varying electron number density, as
happens to a neutrino created in the core of the Sun (Fig. 10.1), the
matter modi®cation to the mass of the electron neutrino can cause a
large neutrino oscillation, even though the vacuum mixing angle is very
small. As the neutrino leaves the Sun, the oscillation is then effectively
frozen, since further oscillations have a small amplitude. It is found that,
13.5 Atmospheric neutrinos
195
if these effects are to account for the solar neutrino de®cit,
m2 †c4 10ÿ5 eV2, sin2 2 10ÿ2 .
13.5
Atmospheric neutrinos
A quite different scale on which to search for neutrino oscillations is given
by atmospheric neutrinos. The Earth is continually bombarded by cosmic
rays, which consist for the most part of high-energy protons and electrons. The protons, in their collision with nuclei in the upper atmosphere,
produce -mesons. The -mesons while still in the upper atmosphere
decay by the chains
‡ !‡ ‡ ! e‡ ‡ e ‡ ÿ !ÿ ‡ ! eÿ ‡ e ‡ The neutrinos and antineutrinos are produced at a mean height
H 20 km, with energies extending to the multi-GeV region.
From the expressions in }13.1 the cross-sections for neutrino and
anti-neutrino scattering increase with energy, so that the detection of
these uncharged leptons becomes easier at higher energy. In water detectors such as SuperKamiokande charged leptons are produced through
reactions essentially of the form
e ‡ n ! eÿ ‡ p;
‡ n ! ÿ ‡ p;
e ‡ p ! e‡ ‡ n;
‡ p ! ‡ ‡ n
which take place within 22.5 kilotonnes of water.
The charged leptons give Cerenkov radiation which provides information on the energy, direction, and identity of the incident uncharged
lepton. At high energy, the direction of the charged lepton is closely
correlated with that of the incident neutrino or anti-neutrino. Electrons
and muons may be distinguished by characteristics of their Cerenkov
signals. (Electrons are scattered more than muons in their passage
through the water. See }14.2.)
196
Neutrinos
Since neutrinos traverse the Earth almost unimpeded, the angle z of
a neutrino's direction with the local vertical at the detector (the zenith
angle) determines its distance z from its point of production in the upper
atmosphere. To a good approximation z is given (using elementary geometry) by
ÿ1=2
z ˆ R2 cos2 z ‡ 2R H ‡ H 2
ÿR cos z ;
where R ˆ 6380 km is the Earth's mean radius, and H 20 km; z varies
between z ˆ H when z ˆ 0, and z ˆ 2R ‡ H when z ˆ .
In Fig. 13.3, the ratio of observed to expected events is plotted as a
function of cos z for electron-like events and for muon-like events. Data
from SuperKamiokande and Kamiokande has been combined.
In the electron data there is no sign of an oscillation. This is consistent with both of the scenarios we have sketched for solar neutrinos. If
either m2 †c4 10ÿ11 eV2 or m2 †c4 10ÿ5 eV2 , no oscillation would
be expected over a distance of R .
The muon data is quite different. There is a clear suppression for
cos z < ÿ0:2, which suggests an oscillation length for the muon neutrino
comparable to R . In a two-neutrino model we can write
"
2
P ˆ 1 ÿ sin 2 sin
2
z
8:1
R
1 GeV
pc
ý
m2 †c4
10ÿ3 eV2
!#
:
m2 might be the difference in the squared masses of the tau neutrino
and the muon neutrino. There is no sign in the data of the muon neutrino
oscillating into an electron neutrino. Detailed ®ts to the data suggest
m2 †c4 10ÿ3 eV2 , and a large value for sin2 2 .
In neutrino physics, we see that nuclear physics, astrophysics, particle
physics, and indeed cosmology, come together, and present challenging
experimental and theoretical problems.
Problems
13.1 Consider allowed ÿ-decays which have a large energy release E0 (e.g. the
decay of 8B, }10.4). In such decays, the effects of Coulomb corrections
and ®nite lepton masses are small. Show that, neglecting these effects,
(a) the mean life depends on E0 as E0ÿ5 ,
(b) the mean electron energy is E0 =2.
Problems
197
Fig. 13.3 The ratio of observed to expected events plotted as a function of cos z ,
for muon-like and electron-like events. (See Harrison, P. F. et al. (1999), Phys.
Lett. B 458, 79.)
To examine the effect of a ®nite neutrino mass on the energy spectrum, only decays with energy in a small range Ee m c2 at the endpoint Ee E0 are signi®cant. Show that the proportion of such decays is
very small, of order 10 Ee =E0 †3 .
13.2 In the K-capture
7
4 Be
atom† ! 73 Li atom† ‡ ,
with the beryllium source at rest, the recoil energy of the lithium atoms
(mass 6536 MeV/c2 ) was measured to be 55:9 1:0† eV (Davis, R.
(1952), Phys. Rev. 86, 976). The mass difference between the two
198
Neutrinos
atoms is 0.862 MeV/c2 . Show that this experiment implied the neutrino
mass to be less than 160 keV/c2 .
13.3 The cross-sections for neutrino interactions are typically of order
2
G2
E
10ÿ17 fm2 ;
F 4 E2 1 MeV
»c†
where E is the neutrino energy in the centre-of-mass frame.
(a) In the core of a supernova, a neutrino will scatter mostly from neutrons.
Show that, in a core of radius 10 km and nucleon number 2:4 1057 , the
mean free path of a neutrino is
1 MeV 2
175 m:
l
E
(b) Consider scatterings in which a neutrino stays a neutrino (as will usually
be the case for muon and tau neutrinos). Show that the time taken for a
neutrino to diffuse from the centre to the surface of this core is of order
2
E
2 10ÿ3 s:
1 MeV
(Assume the neutrino path is a random walk, so that R=l†2 steps are
needed to diffuse a distance R.)
13.4 For neutrinos in thermal equilibrium, the power emitted per unit area at
a surface is given approximately by a formula similar to the Stefan±
Boltzmann law for photons:
power per unit area = a T 4 ;
2 k4B
.
60 c2 »3
(The factor 3 comes from the three neutrino types, and the factor (7/8)
from the Fermi±Dirac statistics.) Show that the core of a star with radius
10 km and surface temperature T given by
where a ˆ 3 7=8† aphoton and aphoton ˆ
T 4 ˆ T04 eÿt= ;
where kB T0 ˆ 4:2 MeV, ˆ 4:6 s, will radiate a total of 3 1058 MeV
by neutrino emission.
14
The passage of energetic particles through
matter
In this chapter we consider the passage of energetic particles through
matter. Nuclear reactions usually result in the production of such particles: -particles, electrons, photons, nucleons, ®ssion fragments, or whatever. In passing through matter, an energetic particle loses its energy,
ultimately largely into ionisation. The instruments of nuclear physics
are designed to detect and measure this deposited energy, and so it is
upon these processes that our knowledge of nuclear physics rests.
The subject is also basic to an understanding of the biological effects
of energetic particles, since a living cell can be damaged by the ionisation.
This can be of positive bene®t, as in the destruction of malignant tissue in
cancer treatment, or a danger from which, for example, workers in the
nuclear power industry must be shielded. Shielding calculations also
depend on the physical principles set out in this chapter.
We limit the discussion to particles with kinetic energies up to around
10 MeV, in line with the nuclear physics described in Chapters 4±12. It is
intended to give the reader a qualitative comprehension, rather than a
compendium of the most accurate formulae and data available for quantitative work.
14.1
Charged particles
We consider ®rst the passage of charged particles, such as protons and particles, through gases. For charged particles of energy <10 MeV, the
dominant mechanism for energy loss is the excitation or ionisation of the
atoms (or molecules) of the gas: electrons being excited to higher bound
199
200
The passage of energetic particles through matter
energy levels in the atom, or detached completely. The essential physics of
the process may be understood using classical mechanics.
We consider a `fast' particle, charge ze, velocity v, energy E, passing a
particle of charge z 0 e, mass mR , initially at rest. We suppose that the fast
particle deviates a negligible amount from its initial straight-line path
along the x-axis (Fig. 14.1), and the rest particle at the point 0; b; 0†
moves only a negligible distance during the encounter. The distance b is
called the impact parameter.
The equation of motion of the fast particle is
dp
ˆ zeE;
dt
where p is its momentum and E is the electric ®eld due to the `rest'
particle. The magnetic ®eld due to the `rest' particle will be negligible.
This equation remains valid for relativistic momenta.
The ®eld E has components
Ex ˆ
1
z 0 e†x
;
4"0 b2 ‡ x2 †32
Ey ˆ ÿ
1
z 0 e†b
:
4"0 b2 ‡ x2 †32
Thus the change in momentum of the fast particle along its direction of
motion is small, for if we approximate its motion by x ˆ vt,
Z
ÿ
zz 0 e2
Ex dt px ˆ ze
4"0
ÿ1
1
!Z
1
ÿ1 b2
vt dt
3
‡ v2 t2 †2
ˆ 0;
whereas the particle acquires transverse momentum pT ˆ py given by
Fig. 14.1 The approximate trajectory of a `fast' particle passing a `rest' particle.
14.1 Charged particles
201
ÿ
!Z
ÿ
!
1
zz 0 e2
b dt
zz 0 e2 2
:
pT ˆ ze
Ey dt ˆ ÿ
ˆÿ
4"0 ÿ1 b2 ‡ v2 t2 †32
4"0 bv
ÿ1
Z
1
14:1†
(The integral is easily evaluated by the substitution vt ˆ b tan .)
Since momentum is conserved overall, the `rest' particle acquires
momentum ÿpT † and, assuming that it does not attain a relativistic
velocity, gains kinetic energy p2T =2mR †. This energy must be lost by the
fast particle:
ÿ
!2
p2T
zz 0 e2
1
ˆ ÿ2
:
E ˆ ÿ
2
2mR
4"0 b v2 mR
14:2†
Note that E does not depend on the mass of the fast particle, and that
the calculation is valid for relativistic velocities of the fast particle.
In applying this result, the `rest' particles are the atomic nuclei and
atomic electrons of the gas. For an atomic nucleus of atomic number Z,
z 0 ˆ Z, and (except for hydrogen) mR 2Zmp . For an electron z 0 ˆ ÿ1
and mR ˆ me . Using the formula (14.2), when a fast charged particle
passes through a gas the ratio of the energy lost to the atomic electrons,
to the energy lost to the atomic nuclei, is 2mp =me 4 103 (since each
nucleus has Z electrons). Thus the energy lost to the nuclei is negligible
compared with that lost to the electrons, and we shall only take the latter
into account. (We are implicitly assuming that the velocity of the fast
particle is large compared with the velocities of the atomic electrons in
their orbits.)
If the gas is of mass density , and consists of atoms of atomic
number Z, atomic mass ma , the number of electrons per unit volume is
=ma †Z. When the fast particle moves through a distance dx in the gas it
passes, on average, =ma †Z 2b db dx electrons with impact parameter
between b and b ‡ db, and the energy lost to these electrons is
ÿ
ze2
d E ˆ ÿ4
4"0
2
!2
Z 1 db
dx:
ma v2 me b
Integrating this expression over all impact parameters between bmin and
bmax , the total rate of energy loss along the path, or stopping power, is
202
The passage of energetic particles through matter
ÿ
!2
dE
ze2
Z 1
ÿ
L;
ˆ 4
4"0 ma me v2
dx
14:3†
where L ˆ ln bmax =bmin †.
Since ma A atomic mass units, where A is the mass number of the
atoms, we write this as
ÿ
dE
Z
zc 2
ˆD
L;
dx
A
v
14:4†
where
ÿ
e2
D ˆ 4
4"0
!2
1
ˆ 0:307 MeV cm2 gÿ1 :
me 931:5 MeV†
and the mass density of the material is expressed in g cmÿ3 . (Note the
units.)
We have introduced parameters bmax and bmin . Our formula (14.2)
clearly breaks down for small b, since the energy transfer cannot be
inde®nitely large; it also breaks down at large b, since to ionise the
atom the energy transfer cannot be inde®nitely small. A quantum
mechanical calculation by Bethe which holds for charged particles
other than electrons and positrons gives equation (14.4) with
!
" ÿ
#
2ÿ 2 me v2
v2
L ˆ ln
ÿ 2 ;
hIi
c
14:5†
where hIi is a suitably de®ned average ionisation energy over atomic
electrons.
The form of (14.5) can be understood qualitatively from the following
considerations. In quantum mechanics a particle is represented by a
wave-packet, and for the classical treatment to be a good approximation
the dimensions of the wave-packet x must surely be less than the impact
parameter b. By the uncertainty principle, the minimum size of a wavepacket is »=p ˆ »= ÿmv†. (For a particle of mass m moving relativistically
1
we must include ÿ ˆ 1 ÿ v2 =c2 †ÿ2 .) In the centre-of-mass frame of the
two particles, the uncertainty in position is the same for both. If the fast
particle is massive compared with an electron, then in the centre-of-
14.1 Charged particles
203
momentum frame it is nearly at rest and the electron has velocity v and
hence momentum p ÿme v. This suggests we should in this case take
»
:
ÿme v
bmin At large impact parameters the energy transfer is small, and we must
recognise that the electrons are bound in atoms and have discrete energy
levels, so that there will be a minimum energy of excitation, of the order
of the ionisation energy I of the electron. In a `collision time' c the energy
of a particle can be uncertain by E ˆ »=c , so that to transfer energy I
requires c < »=I. In our case the collision time c b=ÿv, where the
factor ÿ comes from relativistic considerations, so we take
bmax ˆ
»ÿv
:
I
We have then
ÿ
!
bmax
ÿ 2 me v2
:
ln
L ˆ ln
bmin
hIi
This expression is very similar to (14.5). hIi is usually treated as a parameter and determined by ®tting the formula to experimental data.
Though the formula is derived for gases, it is used for liquids and crystals
by adjusting hIi. For compounds `Bragg's additivity rule' is found to hold
quite well: the energy losses computed for each constituent separately
may simply be added.
The energy loss increases as the particle slows down, due to the 1=v2
factor. (The logarithm is only a slowly varying function of its argument.)
Hence the heaviest ionisation is found towards the end of the track. This
is indeed the case, but the formula was derived for fast particles. At low
velocities, of the order of the atomic electron velocities, it becomes invalid, and ÿdE=dx† decreases to zero as the particle comes to rest. Figure
14.2 plots the stopping power for protons in copper from experimentally
determined data, together with the Bethe formula (14.5) with a ®tted
value of hIi.
The energy-loss equation is of the form
204
The passage of energetic particles through matter
Fig. 14.2 The stopping power for protons and deuterons in copper. The theoretical curve is from equation (14.5) with hIi ˆ 0:375 keV. The experimental points
are from Andersen, H. H. et al. (1966), Phys. Rev. 153, 338.
ÿ
dE
dT
ˆÿ
ˆ z2 function of v=c†;
dx
dx
where T is the kinetic energy of the fast particle. In relativistic mechanics,
T is given by
T ˆp
Mc2
ÿ Mc2 ;
1 ÿ v2 =c2 †
so that we can express v=c† as a function of T=Mc2 †, and write also
ÿ
dT
ˆ z2 F T=Mc2 †;
dx
14:6†
where M is the mass of the fast particle. We can use this result to scale the
data for, say, protons, to apply to other particles. For example, the
14.1 Charged particles
205
stopping power of copper for a 2.5 MeV proton is that for a 5 MeV
deuteron, since md 2mp and z ˆ 1 for both. The validity of this scaling
is exempli®ed by the experimental data also shown in Fig. 14.2.
The result (14.5) holds for particles massive compared with an electron, and must be modi®ed for fast electrons or positrons passing
through matter. In particular, the momentum of either electron in the
p
centre-of-mass system is ÿme v= ‰2 ÿ ‡ 1†Š, so that bmin becomes
p
» ‰2 ÿ ‡ 1†Š=ÿme v. There are other quantum corrections, and the
expressions for electrons and positrons differ slightly. However, another
energy-loss mechanism becomes signi®cant for electrons and positrons
at the higher energies in our range. This is Bremsstrahlung or, in
English, `braking radiation', which is the energy loss by emission of
electromagnetic radiation when a charged particle accelerates and
decelerates during its collisions with the constituent atoms of the matter
it is passing through. We shall not treat this in detail. Bremsstrahlung is
most signi®cant in heavy elements, in which the Coulomb ®elds of the
nuclei are strongest. For electrons and positrons, the ratio of energy loss
rates is given approximately by
Bremsstrahlung energy loss T Z ‡ 1:2†
;
ionisation energy loss
700
where T is the kinetic energy in MeV, and Z the atomic number of the
material.
The range R T0 † of a fast particle of initial kinetic energy T0 , mass M,
is the mean distance it travels before it stops. Using the energy-loss equation in the form (14.6),
Z
0
dT
1
ˆ 2
R T0 † ˆ
dT=dx
z
T0
Z
T0
0
dT
Mc2
ˆ
F T= Mc2 ††
z2
Z
T0 =Mc2
0
du
:
F u†
Another useful scaling law follows from this equation (Problem 14.1).
Given the expression for F T= Mc2 †† the integral can be performed
numerically. If a constant `mean' L is used in the formula for dT=dx
we obtain the approximate result (Problem 14.2)
ÿ
!
A=Z†
T2
:
R T† Dz2 L Mc2 ‡ T
14:7†
206
The passage of energetic particles through matter
Note the mass M appearing in the denominator. If Bremsstrahlung is
negligible, a similar result holds for electrons, with L of the same order
of magnitude for the same material. Thus it is clear that in a given
material electrons and positrons travel a much greater distance on average than protons or other charged particles of the same kinetic energy. A
positron may annihilate with an electron whilst still in motion, but even in
lead, where electrons are abundant, the probability that it comes to rest
before it annihilates is greater than 80%.
In connection with energy loss, it is often of interest to know the total
amount of ionisation caused by the deposition of energy. The primary
electrons knocked out of atoms may have suf®cient energy to ionise
further atoms. It is found experimentally that the total number of electron±ion pairs produced is closely proportional to the energy deposited,
independently of the charge and velocity of the fast particle. The average
energy deposited per electron±ion pair formed is, typically, 30±40 eV for
gases, and 3±4 eV for semiconductors.
14.2
Multiple scattering of charged particles
We have assumed that the `fast' particle travels along an approximately
straight path, but it is of course de¯ected in collisions. The angle of
de¯ection in a single collision is given by
ÿ
!
pT
zz 0 e2 2
ˆ
;
p
4"0 bpv
using equation (14.1). This result agrees with the small-angle limit of the
well-known Rutherford scattering formula.
If the de¯ections occur randomly, as in a gas, the mean square transverse momentum after several collisions (i† is given by
p2T ˆ
X
i
piT †2 ;
treating the vectors piT as a random walk.
Since for the atomic nuclei z 0 ˆ Z, and for the atomic electrons
0
z ˆ ÿ1, it is the nuclei rather than the electrons which are responsible
for scattering the fast particle (except in hydrogen) and we can regard the
electrons as simply screening the Coulomb ®eld of the atomic nucleus at
large impact parameters.
14.3 Energetic photons
207
In a distance x, the fast particle passes =ma † 2b db x nuclei
with impact parameters between b and b ‡ db, so that
p2T
ÿ
!2 Z
2 zZe2 4 db
ˆ
x:
ma 4"0 v2
b
Again, we have to impose a maximum and minimum b. We should here
take bmax an atomic dimension, beyond which the electrons screen out
the ®eld of the nucleus, and bmin a nuclear dimension, since for an
energetic particle it is only when the impact parameter becomes comparable with the nuclear size that the Rutherford formula gives large-angle
scattering, and our approximate expression breaks down. Such largeangle scatterings, though historically important (}4.1), are rare. At even
smaller distances the Coulomb ®eld of the nucleus is modi®ed and nuclear
interactions might occur.
R
Hence we have, roughly, db=b ln A =fm† 10, and the mean
squared de¯ection in a path length x is given by
2 2 2
p2T
2 ÿ1
2 x Z z
:
3
cm
g
MeV
†
p2
pv†2 A
14:8†
Thus the effect of multiple scattering increases rapidly with the atomic
number Z of the material. For heavy particles, pv ˆ mv2 ˆ 2T, and for
energetic electrons, pv pc T. Hence the mean square de¯ection per
unit length is not very sensitive to the mass of the particle, at given kinetic
energy. However, the effects of multiple scattering become more evident
for electrons and positrons because of their longer path length, and the
concept of a well-de®ned linear range is not applicable to these particles.
14.3
Energetic photons
An energetic charged particle loses its energy in small fractions as it
passes through matter, and for a given initial energy it travels a fairly
well-de®ned distance before it comes to rest. When an energetic photon
interacts with an atom, it is totally absorbed or scattered. The intensity of
an initially collimated beam of photons is thereby attenuated, while the
individual photons left in the beam are unscathed. It is useful to de®ne an
attenuation coef®cient, in terms of the total cross-section, tot E†, for a
photon of energy E ˆ »! to interact with an atom. We consider a thin
208
The passage of energetic particles through matter
section of the material, of area S, thickness dx, normal to the direction of
the photons. If the material is of density and made up of atoms of mass
ma , the section will contain =ma †S dx atoms. For suf®ciently small dx,
multiple scatterings can be neglected. If n incident photons impinge at
random on the slab, then from the de®nition of cross-section (Appendix
A) the number of photons dn lost from the beam is given by (Fig. 14.3):
dn
area covered by cross-sections
ˆÿ
n
total area
=ma †S dx tot
ˆÿ
ˆ ÿ =ma †tot dx:
S
Integrating this equation yields
n x† ˆ n 0†eÿx ;
where ˆ tot =ma is called the linear attenuation coef®cient. It is usual
to give data in terms of the mass attenuation coef®cient =† ˆ tot =ma ,
as a function of photon energy. The attenuation coef®cient for compounds can be calculated to within a few per cent by assuming
ˆ
X
i
i
i tot
=mia †;
where i labels the ith constituent.
In the range of photon energies from 1 keV to 10 MeV there are three
main contributions to tot : photo-electric absorption in atoms, the
Compton effect, and pair production. The relative importance of these
Fig. 14.3 Effective total cross-section presented to photons incident normally on
a slab of area S, thickness dx.
14.3 Energetic photons
209
contributions varies with energy (Fig. 14.4). These processes do not interfere with each other and we can take
tot ˆ a ‡ ZC ‡ p :
At low energies the photo-electric effect is dominant. In this process
the photon is absorbed completely by the atom, and an atomic electron is
raised to a higher unoccupied bound state or an unbound state. In Fig.
14.4 it will be seen that the cross-section for photo-electric absorption a
rises sharply at the energies corresponding to the onset of ionisation of
the L and K shell electrons. These energies are higher in heavier elements
in which the core electrons are more tightly bound (Fig. 14.5).
At energies above the K shell absorption edge, the photo-electric
absorption falls off and the scattering of photons by electrons takes over
as the main contribution to the attenuation. This is Compton scattering, and
some of the photon energy goes into the recoil energy of the electron. At
Fig. 14.4 Contributions to the mass attenuation coef®cient for photons in lead.
(Data for this ®gure and Fig. 14.5 from Review of Particle Properties (1983), Rev.
Mod. Phys. 56, S1.)
210
The passage of energetic particles through matter
Fig. 14.5 The mass attenuation coef®cient for photons in some elements spanning the periodic table. Note that both the horizontal and vertical scales are
logarithmic.
these energies the atomic binding of the electrons may be neglected and they
may be treated as free. The momentum±energy conservation calculation is
elementary and well known. The calculation of the Compton scattering
cross-section C , like those of a and p , is a well-understood calculation
in quantum electrodynamics, but the order of magnitude of C can be found
from a simple classical calculation which is valid for low energies »! me c2 † at which the electron recoil is negligible.
In a classical picture, the electron vibrates with the frequency of the
incident electromagnetic wave and emits a secondary wave at the same
frequency. The result of the classical calculation (Problem 14.7) is an
effective cross-section T where
14.4 The relative penetrating power of energetic particles
211
ÿ
!2 8 e2
1 2
T ˆ
3 4"0
me c2
ˆ 0:665 10ÿ28 m2 ˆ 0:665 b:
This limiting value, which does not depend on frequency, is known as the
Thomson scattering cross-section. At higher energies »! 0 me c2 † quantum effects become important, and the Compton scattering cross-section
falls below this value. The corresponding Compton scattering from the
atomic nuclei is reduced by many orders of magnitude because of the
(mass)2 factor in the denominator.
Since there are Z electrons for each atom, the attenuation due to
Compton scattering is given by
C ˆ =ma †ZC Z
:
1 amu A C
For elements other than hydrogen Z=A† 12, so that their plots of =†
approximately coincide in the energy range where Compton scattering
dominates, as Fig. 14.5 shows. In this ®gure the value of =† for hydrogen at low energies is close to T =mH † ˆ 0:397 cm2 gÿ1 .
At photon energies »! > 2me c2 1:02 MeV pair-production becomes
possible (}2.3). This is the process
ÿ ‡ nucleus† ! e‡ ‡ eÿ ‡ nucleus†;
which can occur most readily in the Coulomb ®eld of a heavy nucleus.
The cross-section p increases with energy, and eventually pair-production dominates over other processes. It can be regarded as the inverse
process to Bremsstrahlung, and the cross-section p increases with Z
similarly. This is why the turn-up in the curves of Fig. 14.5 is most
pronounced in the case of lead.
14.4
The relative penetrating power of energetic particles
Table 14.1 sets out the ranges of -particles and electrons of 1 MeV
energy, and the attenuation length, ÿ1 , of photons of 1 MeV energy,
in air and in soft tissue.
The high penetrating power of energetic photons (called X-rays or ÿrays, depending on whether they come from atomic or nuclear processes!)
212
The passage of energetic particles through matter
Table 14.1. Ionising path lengths for 1 MeV electrons and 1 MeV
-particles, and 1 MeV photon attenuation lengths, in air and in soft
tissue
Electron
Alpha particle
Photon
Air (cm)
Soft tissue (cm)
380
0.52
1:1 104
0.43
7 10ÿ4
14
(Data from American Institute of Physics Handbook, 3rd ed. 1972, New York:
McGraw-Hill.)
is used in medical diagnosis, and in industry, for imaging. The familiar
`X-ray photograph' depends on differences in X-ray absorption in different materials.
The short range of electrons in matter is also exploited. A source of þactivity of appropriate energy, provided that it is not a signi®cant source
of secondary ÿ emission (see }7.5), may be implanted in diseased tissue to
give a localised source of radiation, so that diseased tissue is destroyed
whilst neighbouring healthy tissue is unaffected. 32P is an example of such
a clean þ source.
In cases of accidental exposure to radiation, sources of -radiation
and þ-radiation are usually only harmful if taken into the body: because
of the short ranges involved, external sources are effectively shielded from
the body by any intervening material.
The analysis of the effect of external X-rays or ÿ-rays is more complex. The attenuation length for a photon is the mean distance it travels
before depositing any ionising energy. In the 1 MeV region, where
Compton scattering predominates, the recoil electron from the scattering
produces ionisation, but the scattered photon can still have suf®cient
energy to undergo further scattering and produce more ionisations until
its energy becomes so low that photo-electric absorption takes place. The
situation is best analysed by computer simulations, using so-called
Monte-Carlo techniques.
Problems
14.1 Show that if Rp T† is the range of a proton of kinetic energy T, the range
RM TM † of a charged particle of mass M, kinetic energy TM , and charge
ze is given by
Problems
RM TM † ˆ
213
M
Rp mp TM =M†:
z2 mp
show that the integral
14.2 If L in equation (14.4) is replaced by a constant L,
for the range of an ionising particle can be evaluated to give the approximate result (14.7).
14.3 For `back-of-envelope' calculations, a useful estimate of the mean ionisation energy hIi for an atom of atomic number Z is hIi ˆ 12Z eV.
Show that for -particles of kinetic energy 2 MeV in aluminium the L
of equation (14.5) 2; for electrons of kinetic energy 2 MeV in aluminium, L 10. Use these values to estimate the range of 5 MeV -particles and of 5 MeV electrons in aluminium (mass density 2:7 g cmÿ3 ).
14.4 Show that for a non-relativistic particle of mass M, velocity v,
dE=dx† ˆ M dv=dt†. Replacing L by a constant L in equation (14.4),
show that the time for a non-relativistic particle with initial velocity v0 to
come to rest is (4/3) (range)/v0 .
Estimate the time taken by the -particle of Problem 14.3 to come to
rest.
14.5 In a neutron detector of the type described in Problem 8.2, estimate
roughly the number of ion pairs produced in the helium gas per neutron
interaction and the distance over which the ionisation is deposited.
14.6 50 keV X-rays are in common use in dentistry. Estimate the thickness of
lead sheet (density 11:4 g cmÿ3 ) that will absorb 99.9% of such radiation
at normal incidence.
14.7 Larmor's formula for the power P radiated from a non-relativistic particle of charge e and acceleration a is
ÿ
!
2 e2 a2
:
Pˆ
3 4"0 c3
Show that classically an electron in an electric ®eld E ˆ E0 cos !t will
radiate energy at a mean rate
ÿ
!
1 e2 jE0 j2 e2
:
Pˆ
3 m3e c3 4"0
The incident energy ¯ux in a plane electromagnetic wave is c"0 jE0 j2 =2
(cf. }12.8). Hence obtain the Thomson scattering formula.
15
Radiation and life
Life on Earth has evolved and is sustained by the light and heat of the
Sun. In addition to this essential and almost entirely benign ¯ux of electromagnetic energy, living organisms have always been subject to the
hazards of natural ionising radiation. In the twentieth century man's
activities added somewhat to these hazards. On the other hand, ionising
radiation is used to great advantage in industry and for diagnostic and
therapeutic purposes in medicine, and nuclear power is not without its
bene®ts. The interaction between ionising radiation and living tissue is
therefore a matter of great interest and importance.
15.1
Ionising radiation and biological damage
The basic unit of living tissue is the cell. Cells are complex structures
enclosed by a surface membrane. A cell has a central nucleus. This contains DNA (deoxyribonucleic acid) molecules, which code the structure,
function, and replication of the cell. The famous `double helix' of the
DNA molecule has a diameter of about 2 nm. About 80% of a cell
consists of water.
The induction of cancer or of hereditary disease by low levels of
ionising radiation is believed to be related to damage to the DNA molecules. This can happen by direct ionisation of the molecule, or indirectly
through ionisation of the water molecules in the cell. The break-up of a
water molecule may produce a hydroxyl (OH)ÿ ion that is highly reactive
chemically and may attack the DNA molecule.
A single broken strand of DNA is rapidly repaired (within hours) by
cellular enzyme systems, the unbroken strand of the DNA acting as
214
15.2 Becquerels (and curies)
215
template. However, if there is at the same time adjacent damage to the
other strand, neither can be repaired in this way. There may then be
errors in the repair processes, and consequent abnormalities in cell behaviour. The cell may, for example, die, which at low levels of radiation is
usually a matter of little consequence. Damage to a cell which causes later
uncontrolled cell division may lead ultimately to a tumour developing,
albeit usually after a long period of latency. A cell involved in reproduction which is damaged but survives may transmit genetic defects to subsequent generations.
The relative biological damage caused by different types of radiation
can be understood in terms of their effectiveness in causing a double
break in DNA strands. For example, we saw in Chapter 14 that electrons
and positrons travel a much greater distance in a given material than particles of the same energy, and produce roughly the same number of
electron±ion pairs. Thus the -particle ionisations are more closely
spaced, and it is more likely that an -particle will damage both strands
of a DNA molecule, compared with a þ-particle of the same energy.
15.2
Becquerels (and curies)
It will be helpful at this point to introduce some specialised units into our
vocabulary.
Radioactive nuclei may emit -particles, electrons, positrons,
photons, or ®ssion products. The activity of a given nuclear species in a
given sample is the average number of decays per second of that species,
and is measured in becquerels: 1 Bq corresponds to an average of one
decay per second.
The total activity of a newly prepared sample may initially increase
with time, since the daughter products of a radioactive nucleus may also
be radioactive (Problem 15.2), though ultimately the total activity must
decay to zero.
The becquerel is the SI unit which has replaced the curie:
1 Ci ˆ 3:7 1010 Bq. The curie was de®ned originally as the 226Ra activity of a source containing 1 g of 226Ra. Since the mean life of 226Ra is
7:28 1010 s, it is easy to check that the de®nition above is approximately
consistent with the older de®nition.
216
Radiation and life
15.3
Grays and sieverts (and rads and rems)
The absolute absorbed dose of radiation at any point in a material is
de®ned as the energy per unit volume that has been absorbed by the
material, divided by the mass density at the point. The SI unit for
absorbed dose is the gray, corresponding to one joule of absorbed energy
per kilogram of material. 1 Gy ˆ 6:24 1012 MeV kgÿ1 :† An older unit
is the rad: 1 Gy ˆ 102 rad. In practice, a quoted absorbed dose will be an
average over some region, for example a whole body average or an average over some particular organ of the body.
It has been found that radiation damage to living tissue is not simply
proportional to the absolute absorbed dose, but depends on several other
factors, of which one is the type of radiation. For example, for the same
number of grays, -particles are more damaging than ÿ-radiation. From
medical experience, different types of radiation have been given radiation
weighting factors wR . The wR factors are dimensionless numbers. For
many purposes it is conventional to take these factors to be
1 for X-rays, ÿ-rays, þ-particles and muons,
5 for protons >2 MeV,
20 for -particles.
Neutrons are uncharged and hence are not directly ionising.
However, in elastic collisions of neutrons with nuclei, the nuclei are set
in motion and become ionising. Neutron capture with ÿ-ray emission,
and nuclear ®ssion, are other possible processes which lead to ionisation.
The radiation weighting factor for neutrons has been found to be strongly
energy dependent and is taken to be
5
10
20
10
5
for
for
for
for
for
<10 keV,
10±100 keV,
100 keV to 2 MeV,
2±20 MeV,
>20 MeV.
The sievert (Sv) is a unit combining the wR factor with the absorbed
dose: the equivalent dose in Sv equals the absorbed dose in Gy, multiplied
by the wR factor for the radiation involved. The equivalent dose in Sv is
an indicator of the potential harm to living tissue of a given dose of
radiation. In practice equivalent doses are usually quoted in millisieverts.
The rem is related to the rad in the same way that the sievert is related to
the gray, so that 1 Sv ˆ 102 rem.
15.4 Natural levels of radiation
217
As an example, the equivalent dose delivered by a 5 MeV -decay in
the body is 20 5 ˆ 100 times the equivalent dose delivered by a 1 MeV
þ-decay in the body.
Different organs and tissues of the human body (liver, bone marrow,
skin, etc.) have different sensitivities to ionising radiation. An effective
dose may be de®ned, weighting the equivalent dose received by the various major organs and tissues by an empirical factor related to the susceptibility to biological damage of these organs and tissues, and summing
over the whole body. This gives a crude but useful `single number' measure of radiation damage. In the rest of this chapter, effective dose is
abbreviated to dose.
15.4
Natural levels of radiation
There are three principal natural sources of ionising radiation: cosmic
rays, radioactive nuclei which participate in the chemistry of the body,
and radioactive elements present in rocks and soil.
Cosmic rays are very high energy particles which permeate the
Galaxy. Those which strike the Earth's atmosphere cause showers of
secondary particles; at sea level these secondaries deliver a dose of
about 0.25 mSv per year to the human body (Problem 15.3). The precise
dose depends on latitude and increases with altitude. At a height of
4000 m the dose would be about 2 mSv per year. Air travel adds an
average of 0.01 mSv per year to the UK cosmic ray dose.
The most signi®cant radioactive nucleus that is found in the body is
40
K. Potassium enters the body with a normal diet, and accounts for
about 0.2% of total body weight. The isotope 40
19 K, which has spin and
parity 4ÿ , has a long mean life of 1:85 109 years, and that which
remains since the Earth's formation constitutes 0.0117% of natural potassium. It is an odd±odd nucleus and can undergo all three types of þdecay, but the most common mode (89%) is electron emission with a
kinetic energy release of 1.32 MeV; the remaining 11% of decays are
mostly by electron capture to an excited state of 40Ar, which then itself
decays by emitting a 1.46 MeV ÿ-ray. From these decays the body
receives a dose of 0.17 mSv per year. Other radioactive nuclei in the
body give in total a contribution of similar magnitude. (This is excluding
the contribution from inhaled radon described below.)
The dose of ÿ-radiation arising from the decay products of radioactive elements in the ground, principally from uranium and thorium,
depends on the local geology and is far more variable. Typically the ÿ-
218
Radiation and life
radiation dose is between 0.2 mSv and 0.4 mSv per person per year, but
in areas of granite rock may be several times higher. A greater hazard can
arise from the inhalation of the isotopes 222Rn and 220Rn of the inert gas
radon. These are decay products of uranium and thorium and being
gaseous can diffuse out into the air. In particular they may emanate
from some building materials, and accumulate in ill-ventilated rooms.
222
Rn decays to a sequence of -emitters (Table 6.1) which are solids
and remain deposited in the lungs. 220Rn, arising from the 232Th chain,
is similarly damaging. The dose received depends on building materials
and construction, subsoil, and ventilation, and obviously varies widely; it
has been estimated that the dose averaged over the UK is about 1.0 mSv
per person per year.
The average natural background radiation thus totals around
2.2 mSv per person per year.
15.5
Man-made sources of radiation
To the natural background radiation dose we must add the dose resulting
from man's activities since the early twentieth century. The most signi®cant contribution to this comes from the medical applications of ionising
radiation in diagnostic radiology and radiotherapy. There are, of course,
very wide variations in the dose an individual receives. The dose from a
chest X-ray is about 0.2 mSv, while someone given a computed tomography scan might receive 10 mSv. Averaged over the UK, the dose per
person from medical applications is about 0.37 mSv per year.
The average dose due to the radioactive fallout from nuclear weapons
testing in the atmosphere in 1999 was 0.004 mSv per year in the UK,
compared with a peak of 0.014 mSv per year in 1963. The average dose
due to the Chernobyl reactor accident in 1986 has declined to 0.001 mSv
per year averaged over the UK, though there are considerable regional
variations. In normal operation the nuclear power industry does not add
appreciably to the average dose. The average dose from all stages of the
nuclear fuel cycle averages to 0.0002 mSv per year. Again there are wide
variations: it has been estimated that people living near some nuclear
facilities receive annual doses of 0.5 mSv.
Many individuals, through their work in medicine or in nuclearrelated (and other) industries, are habitually exposed to higher levels of
radiation than the average. It is necessary to monitor and protect these
people in so far as knowledge will allow. The National Radiation
Protection Board in the UK recommends a maximum exposure of
15.6 Risk assessment
219
15 mSv per year for such workers. Averaged over the entire population,
the average dose arising from occupational exposure is 0.007 mSv.
Thus the average dose per person from arti®cial sources of ionising
radiation is around 0.4 mSv per year in the UK. This is to be compared
with a natural background of 2.2 mSv per year.
15.6
Risk assessment
The gray and sievert are large units in terms of biological damage: whole
body doses of ÿ-radiation between 2.5 Gy and 3.0 Gy given over a short
period are likely to result in a 50% mortality rate within 30 days, in the
absence of medical intervention. At very low levels it is not yet established
with certainty whether or not a threshold for biological damage exists.
There is no way of identifying a cancer induced by ionising radiation from
other cancers of the same type which have appeared spontaneously. At a
low dose rate the number of radiation induced cancers is not statistically
signi®cant, so that extrapolation from data at high doses, where the
effects are evident, is the only way to make an estimate. Risks are usually
assessed on the assumpton of a proportionality between dose and effects,
but the extrapolation from high doses is not straightforward. At low
doses, a double break in DNA strands is likely to come from two distinct
tracks independently causing breaks in the two DNA strands at nearly the
same place in the molecule, with the second break occurring before the
®rst break has been repaired. Such a process may be expected to happen
with a probability proportional to the square of the dose.
Data for whole body exposure to ÿ-radiation come mainly from
studies of survivors of the atomic bombs dropped on Hiroshima and
Nagasaki in 1945, who were exposed to high and uncontrolled doses
for a short period. Information on the hazards of radon and its decay
products comes from studies of miners exposed to high concentrations of
radon. Other information comes from patients who have undergone
radiation treatment for medical reasons.
The International Commission on Radiological Protection has concluded from these studies that, averaged over a `representative' population, the lifetime risk of contracting fatal cancer from unit cumulative
dose of radiation is about 5 10ÿ2 Svÿ1 , and averaged over the working
population is somewhat lower: 4 10ÿ2 Svÿ1 (since this latter average
excludes younger people, who have more years at risk). The emphasis
on fatal cancers allows comparisons to be made with other fatality rates.
For example, a radiation worker occupationally exposed to 1.5 mSv per
220
Radiation and life
year has an additional annual risk of death from radiation induced cancer
of 4 10ÿ2 1:5 10ÿ3 ˆ 6 10ÿ5 , or 1 in 17 000, which is about the
same as the fatality rate from accidents to workers in the UK construction industry. The annual risk of contracting fatal cancer induced by
natural radiation is 5 10ÿ2 2:2 10ÿ3 ˆ 1:1 10ÿ4 , or 1 in 9000.
Risk assessment is important as a guide to action. For example, about
half of the average natural radiation dose comes from radon in buildings.
It is clearly desirable to reduce the risk of those exposed to very much
higher than average radon concentration in their homes or workplaces, as
a consequence of local geology. Quite simple measures (fans, sealing) can
reduce the ¯ow of radon gas released from the soil and from construction
materials.
(The numerical data quoted in this chapter have been taken from
Living with Radiation, National Radiation Protection Board, 1998.)
Problems
15.1
5.9% of all 235U ®ssions produce a 137Cs nucleus within about 5 minutes. The mean life of 137Cs is 44 years. It is a particularly dangerous
radioactive isotope if released in the atmosphere.
Estimate the activity of 137Cs in a reactor that has been running at 3 GW
thermal power for one year. In the Chernobyl accident 13% of this
isotope was released. Estimate its mean activity per square metre if it
was spread over a million square kilometres.
15.2 The mean life of 226
88 Ra (2300 years) is so long that the radium activity of
a newly prepared one curie source will be essentially constant. The mean
life of its daughter nucleus 222
86 Rn is ˆ 5:52 days. Show that the radon
activity approaches the radium activity according to
Rn activity = 1 ÿ eÿt= † Ci.
The subsequent decays in the chain (see Table 6.1) down to 210
82 Pb all
have mean lives of less than an hour, but 210
with a
Pb
is
relatively
stable
82
mean life of 30 years. Estimate the total activity of the source one month
after preparation.
15.3 At sea level most of the ¯ux of ionising particles induced by cosmic rays
consists of muons, and over all angles the total ¯ux is about 170 particles
mÿ2 sÿ1 . The mean muon energy is about 2 103 MeV. In the body the
muons will lose energy predominantly by ionisation. Estimate the
annual body dose due to this source, taking the mean L of equation
(14.4) to be 14.
Problems
221
15.4 The body contains about 18% by weight of carbon, of which a small
proportion is the þ-unstable isotope 146 C. About one-third of the decay
energy of 0.156 MeV is taken by the electron, and there are no associated ÿ-rays. The activity of 1 g of natural carbon is 15.3 decays per
minute. Estimate the annual whole body dose of radiation from this
source.
15.5 Check that the quoted value of 0.17 mSv per year body dose from 40K is
consistent with the information given. (Assume that about half of the
1.32 MeV energy release in eÿ emission is taken by the anti-neutrino,
and take the attenuation length of a 1.45 MeV ÿ-ray in the body to be
17 cm.)
Appendix A
Cross-sections
We begin this appendix by considering neutron cross-sections. There is
some simpli®cation in the case of neutrons, since they are electrically
neutral and do not interact through the long-range Coulomb force. To
a good approximation they can be considered to interact only through the
short-range nuclear force. The concepts developed for neutrons may be
applied almost immediately to other electrical neutral particles, such as
photons. We then turn to the case of charged particles.
A.1
Neutron and photon cross-sections
We consider a neutron approaching from a distance a nucleus which is at
rest. (Any interactions between the neutron and the atomic electrons will
be neglected.) We suppose that, if the nucleus were not present, the probability of the neutron passing anywhere through a circle of radius a,
centred on the nucleus and perpendicular to the direction of the neutron's
motion, would be uniform (Fig. A.1), i.e. the probability of it passing
through an area A would be A= a2 †. We can think of the neutron as a
classical particle, or better, as a quantum-mechanical wave-packet. The
radius a must be large compared with both the size of the wave-packet
and the size of the nucleus. With the nucleus present, an interaction can
take place, for example scattering, induced ®ssion, or radiative capture. It
is found that, provided a is large enough, the probability of an interaction
is inversely proportional to the area a2 , i.e.
probability of interaction =
222
tot
:
a2
A:1†
Cross-sections
223
Fig. A1 Neutron incident on area centred on nucleus at rest.
The constant of proportionality introduced here, tot , is called the total
cross-section. Clearly tot has the dimensions of area. It can be regarded as
the effective area presented to the neutron by the nucleus, but it must be
realised that the cross-section is a joint property of the neutron and
nucleus, and for a given nucleus it is a sensitive function of neutron
energy. The probability of interaction is a quantum-mechanical property:
tot can be very much larger than the geometrical cross-section of the
nucleus.
The system of a moving particle incident on a nucleus at rest is called
the laboratory system. With ®xed targets, it is the situation most easy to
simulate in the laboratory. It is also useful to consider interactions in the
frame of reference in which the nucleus has momentum equal in magnitude but opposite in direction to the neutron. It is clear from the de®nition that the total cross-section is the same viewed in this `centre-of-mass'
system as in the laboratory system.
There will usually be several possible reaction channels, i.e. types of
interaction that can occur. Examples are:
Elastic scattering: the incoming neutron changes direction but, in the
centre-of-mass system, loses no energy.
Inelastic scattering: the incoming neutron changes direction and, even in
the centre-of-mass system, loses energy in exciting the nucleus.
Radiative capture: the incoming neutron is captured by the nucleus. The
resulting nucleus is formed in an excited state, which eventually decays
by photon emission.
Given that a reaction occurs, each reaction channel `i' has a de®nite
probability pi , where
X
i
pi ˆ 1:
224
Appendix A
The partial cross-section, i for the ith channel, is de®ned to be
i ˆ pi tot , and may be regarded as the effective area presented by the
target nucleus to the neutron for that particular reaction. We have
tot ˆ
X
i
i :
Photon cross-sections can be de®ned in direct analogy with neutron
cross-sections, but since photons interact with atomic electrons as well as
with nuclei it is more appropriate to consider the target to be an atom.
Various contributions to the total cross-section for a photon to interact
with an atom are discussed in Chapter 14.
A.2
Differential cross-sections
In considering a particular reaction channel it is often useful to subdivide
it further. For example, in the elastic scattering of neutrons it can be of
interest to know the probability distribution of the angle at which the
neutron emerges from the interaction. Given that an elastic scattering
occurs, if pe ; †dý is the probability that the neutron is scattered into
a small solid angle dý ˆ sin d d, at a polar angle and azimuthal
angle with respect to its incident direction, we write (Fig. A2)
1 de
dý:
pe ; † dý ˆ
e dý
A:2†
Fig. A2 Geometry of elastic scattering from a ®xed target into a small solid
angle dý ˆ sin d d about a polar angle and azimuthal angle .
Cross-sections
225
This de®nes the elastic differential cross-section del =dý, and
Z
de
dý ˆ e :
dý
A:3†
Differential cross-sections are usually measured in the laboratory with
respect to a ®xed target. In the centre-of-mass frame the direction of a
scattered neutron, and hence the angular dependence of the cross-section,
will be different. The kinematic transformation between the frames is
straightforward, and experimental data is often presented in the centreof-mass frame to facilitate comparison with theory.
A.3
Reaction rates
Consider a broad collimated beam of mono-energetic neutrons. Let n be
the number density of neutrons in the beam, and v the neutron velocity.
The neutron ¯ux, i.e. the number of neutrons crossing a unit area normal
to the beam per unit time, is n v. Hence in time dt, the number of
neutrons passing through a circle of radius a centred on a nucleus is
n v dt a2 . From (A.1), the probability of a reaction with the nucleus
taking place in the time interval t, t ‡ dt, given the nucleus is in its ground
state at time t, is n vtot dt. Thus the reaction rate per nucleus is n vtot or
reaction rate = flux cross-section:
We may also consider a single neutron, moving with velocity v in a
random array of nuclei of number density nuc . By the same argument (in
the frame in which the neutron is at rest and the nuclei are regarded as a
beam) the reaction rate is nuc vtot . Given that the neutron exists at t ˆ 0,
the mean time before an interaction takes place is, therefore,
ˆ nuc vtot †ÿ1 , and the mean free path l, the distance it travels in this
time, is
l ˆ v ˆ 1= nuc tot †:
A:4†
(We have assumed that is very much shorter than the quarter of an hour
intrinsic mean life of the neutron.)
226
A.4
Appendix A
Charged particle cross-sections: Rutherford scattering
The dif®culties that arise in charged particle scattering stem from the long
range of the Coulomb force. In }14.2 it is shown that a charged particle,
say a proton, passing a ®xed target, say a nucleus of charge Ze, is
de¯ected through an angle given approximately by
ÿ
!
Ze2
2
;
ˆ
4"0 bpv
A:5†
where p is the momentum and v the velocity of the proton, and b is the
impact parameter, the distance at which the proton would pass the
nucleus if there were no interaction (Fig. 14.1).
Impact parameters between b and b ‡ db correspond to scattering
angles between and ÿ d where
ÿ
!
db 2Ze2
d ˆ 2
:
b 4"0 pv
The effective area presented to the proton which corresponds to this
range db of impact parameters is 2b db, and we can interpret this as a
contribution to the elastic scattering cross-section,
de ˆ 2b db
ÿ
!2
2Ze2
d
;
ˆ 2
4"0 pv 3
using (A.5):
The differential scattering cross-section for small angles is therefore
ÿ
!2
de
1 de
2Ze2
1
;
ˆ
ˆ
dý 2 sin d
4"0 pv 4
where we have replaced sin by . This is the small-angle limit of the
famous Rutherford scattering formula. The same expression is obtained
from a quantum-mechanical calculation.
The differential cross-section becomes very large when is very small,
and the total elastic cross-section, de®ned by the integral (A.3), and hence
the total cross-section, is in®nite. Physically this is because the Coulomb
force is still felt by the proton no matter how large the impact parameter.
Density of states
227
In practice this formally in®nite result is not a serious dif®culty, since
there is always a limit to the experimentalist's ability to measure smallangle scattering, and if one is interested only in elastic scattering through
angles greater than some minimum angle the cross-section is ®nite.
At large impact parameters the Coulomb force is weak, and can only
give rise to small-angle elastic scattering. The cross-sections for other
possible processes are all ®nite.
Appendix B
Density of states
Consider a particle moving freely inside a cubic box of side L, volume
V ˆ L3 . We take the potential to be zero inside the box, and represent the
walls by in®nite potential barriers. The SchroÈdinger equation for the
particle,
ÿ
»2 2
r ý ˆ Eý;
2m
B:1†
is separable in x; y; z† coordinates, and the solutions must vanish at the
walls which we can take to be the planes x ˆ 0 and x ˆ L, y ˆ 0 and
y ˆ L, z ˆ 0 and z ˆ L. These solutions are easily seen to be standing
waves of the form
ý x; y; z† ˆ constant† sin kx x† sin ky y† sin kz z†;
B:2†
provided that we choose k ˆ kx ; ky ; kz † from the values
nx ;
L
n
kz ˆ z ;
L
kx ˆ
nx ˆ 1; 2; 3; . . . ;
ky ˆ
ny ;
L
ny ˆ 1; 2; 3; . . . ;
nz ˆ 1; 2; 3; . . .
to satisfy the boundary conditions. Negative integer values of nx , ny , nz do
not give new states, since they merely change the sign of the wave-function, and such a phase factor has no physical signi®cance.
Thus the allowed values of k form a cubic lattice of points in the ‡;
‡; ‡† quadrant of `k-space'. Each eigenstate (B.2) corresponds to one
point of the lattice, and counting states is equivalent to counting lattice
points. The spacing between these lattice points is =L†, so that the
228
Appendix B
number of points per unit `volume' in k-space is L=†3 . The number of
lattice points with k ˆ jkj† less than some ®xed value k0 is the number
enclosed within the quadrant of the sphere centred at the origin and of
radius k0 . This number must of course be an integer, but for large values
of k0 it will be approximately given by:
(volume of quadrant of sphere) (density of lattice points)
1 4k30 L 3
V 4k30
ˆ
:
ˆ
8 3
2†3 3
B:3†
The number of points with k lying in the range k0 < k < k0 ‡ dk0 is the
differential of (B.3):
V
4k20 dk0 :
2†3
B:4†
We will consider the case of a spin 12 fermion (for example, an electron
or a nucleon). Then two states (`spin-up' and `spin-down') can be assigned
to each k value, from (B.3) the number of states N 0 with k < k0 is
N0 ˆ 2
V 4k30
;
2†3 3
or
k30 ˆ 32
N0
:
V
B:5†
For the non-relativistic SchroÈdinger equation (B.1), the energy E of a
particle in a state of speci®ed nx ; ny ; nz † and either spin, is related to k by
Eˆ
»2 2
»2 2
kx ‡ k2y ‡ k2z † ˆ
k :
2m
2m
B:6†
The integrated density of states N E† is de®ned as the number of states
1
with energy less than E. From (B.6) k ˆ 2mE= »2 †2 ; hence using (B.5) we
have
3
V 2mE 2
N E† ˆ 2
:
3
»2
B:7†
The density of states n E† ˆ dN =dE, so that n E† dE is the number of
states with energy between E and E ‡ dE:
Density of states
3
dN
V 2m 2 12
ˆ 2
n E† ˆ
E:
dE
2 »2
229
B:8†
If the spin factor is omitted,
3
V 2m 2 12
E:
n E† ˆ 2
4 »2
B:9†
In scattering problems, it is convenient to consider a large volume L3
and impose `periodic boundary conditions' on the wave-functions:
ý x ‡ L; y; z† ˆ ý x; y; z†;
ý x; y ‡ L; z† ˆ ý x; y; z†;
ý x; y; z ‡ L† ˆ ý x; y; z†:
Instead of the standing waves (B.2), the solutions of the wave-equation
consistent with the boundary conditions are the travelling waves
eikr ˆ eikx x eiky y eikz z ;
where, to satisfy the periodicity conditions, we must now take
kx ˆ
2nx ;
L
nx ˆ 0; 1; 2; . . . ;
etc:
The density of points in k-space becomes L=2†3 . However, permutations of sign kx ; ky ; kz † now correspond to different states (travelling waves in different directions), and the lattice points corresponding to
distinct states with jkj < k0 ®ll the whole sphere of radius k0 in k-space.
We thus arrive again at the results (B.3) and (B.5); (B.7) and (B.8), which
hold for non-relativistic spin 12 fermions, are also still valid.
In fact, in the limit when the linear dimensions of the box become
large compared with the de Broglie wavelength of the particle at energy E,
the result for the density of states at energy E becomes independent both
of the boundary conditions imposed and of the shape of the box, provided this remains simple. The integrated density of states in a sphere is
illustrated in Fig. 5.2.
230
Appendix C
Problems
B.1(a) For a single particle in a large volume V, show that the number of
allowed k-values in a small volume d3 k ˆ dkx dky dkz of k-space is
V 3
d k:
2†3
(b) Show that, for two particles (1) and (2), the wave-vector associated with
the centre-of-mass motion is K ˆ k1 ‡ k2 and with the relative motion is
k ˆ m1 k2 ÿ m2 k1 †= m1 ‡ m2 †. Hence show
d3 K d3 k ˆ d3 k1 d3 k2
and that the number of K; k† values with K in the range K, K ‡ dK and
k in the range k, k ‡ dk is V 2 =44 †K 2 dK k2 dk if the particles are distinguishable, but V 2 =84 †K 2 dK k2 dk if the particles are identical.
Appendix C
Angular momentum
Students are referred to texts on quantum mechanics for the derivations
of the results summarised in this appendix, which is intended as no more
than an aide-meÂmoire.
C.1
Orbital angular momentum
In the shell model of both atomic and nuclear physics the single-particle
SchroÈdinger equation, neglecting effects of the intrinsic spin of the particle, is of the form
ÿ
!
»2 2
Hý ˆ ÿ
r ‡ V r† ý r† ˆ Eý r†;
2M
C:1†
where the potential energy V r† is spherically symmetric, a function of the
radial coordinate r only. Because of spherical symmetry, the operator r2
is most useful in spherical polar coordinates r; ; †, in which the
SchroÈdinger equation takes the form
ÿ
!
»2 1 @2
L2
ÿ
rý† ‡
‡ V r† ý ˆ Eý;
2M r @r2
2Mr2
C:2†
Angular momentum
231
where L2 ˆ L2x ‡ L2y ‡ L2z and L is the orbital angular momentum operator,
L ˆ r p ˆ r ÿi »r†:
L acts only on the angular coordinates ; †. For example,
@
@
@
ˆ ÿi » :
Lz ˆ ÿi » x ÿ y
@y
@x
@
From the de®nition of L, it is not dif®cult to obtain the commutation
relations
‰Lx ; Ly Š ˆ i»Lz ;
2
‰Ly ; Lz Š ˆ i »Lx ;
2
‰Lz ; Lx Š ˆ i »Ly ;
2
‰L ; Lx Š ˆ ‰L ; Ly Š ˆ ‰L ; Lz Š ˆ 0:
C:3†
Because Lx , Ly , Lz do not commute, it is not generally possible for a
wave-function to be simultaneously an eigenstate of any two of them, but
it is always possible to construct simultaneous eigenstates of L2 and any
one of Lx , Ly , Lz . It is conventional to choose L2 and Lz .
The simultaneous eigenstates of L2 and Lz are denoted by Ylm ; †,
where
L2 Ylm ˆ l l ‡ 1† »2 Ylm
C:4†
Lz Ylm ˆ m »Ylm :
The allowed values of l are the integers l ˆ 0; 1; 2; 3; . . . and, for a given l,
m takes one of the 2l ‡ 1† values ÿl; ÿl ‡ 1; . . . ; l ÿ 1; l. The functions
Ylm ; † are well-known spherical harmonics, and are normalised so that
Z
Yl0 m 0 Ylm
Z
dý ˆ
0
Z
d sin 2
0
dYl0 m 0 ; †Ylm ; †
ˆ ll 0 mm 0 :
p
For example, Y00 ˆ 1= 4; a state of zero orbital angular momentum is
spherically symmetric. Also
232
Appendix C
r
r
3 x ‡ iy
3
Y11 ˆ ÿ
sin ei
ˆÿ
8 r
8
r
r
3 z
3
cos ˆ
Y10 ˆ
4 r
4
r
r
3 x ÿ iy
3
Y1ÿ1 ˆ
ˆ
sin eÿi :
8 r
8
Note that these states with l ˆ 1 can be formed from the components of
the unit vector x=r; y=r; z=r†.
These examples illustrate a general rule: the parity of a state of given l
is ÿ1†l .
From (C.2), the eigenfunctions of the SchroÈdinger equation are of the
form
ýnlm ˆ unl r†Ylm ; †
C:5†
where unl satis®es the ordinary differential equation
ÿ
!
» 2 1 d2
»2 l ‡ 1†
ru † ‡
‡ V r† unl r† ˆ Enl unl r†:
ÿ
2M r dr2 nl
2Mr2
There are several examples of potentials V r† for which the radial functions unl r† are elementary, and many others for which the numerical
solutions are easy to program on computers.
Note that the energy eigenstates (C.5) are also eigenstates of L2 and
Lz . This is only possible because of the spherical symmetry of V r†, which
allows L2 and Lz (which act on and only) to commute with the energy
operator.
C.2
Intrinsic angular momentum
A particle may have an intrinsic angular momentum or spin s, satisfying
the same commutation relations as (C.3). The eigenvalues of s2 are
s s ‡ 1† »2 , and ms can take 2s ‡ 1† values from ÿs to ‡s. In the case
of orbital angular momentum treated above, l must be a positive integer.
This condition stems from the single-valuedness of the wave-function in
space. The quantum number s is not subject to this restriction, since the
coordinates on which s acts are internal to the particle, and we require
only that 2s ‡ 1† should be an integer. Thus we may have s ˆ 12, as is the
case with leptons and nucleons. For s ˆ 12 there are two eigenstates, cor-
Angular momentum
233
responding to ms ˆ ‡ 12 ; ms ˆ ÿ 12. We may denote these by j ‡ 12i and
j ÿ 12i. A general wave-function for the spin 12 fermion is a superposition
of `spin-up' and `spin-down' states of the form
ý r; ms † ˆ ý‡ r†j ‡ 12i ‡ ýÿ r†j ÿ 12i:
The two independent spin states j ‡ 12i, j ÿ 12i may be represented by
column vectors
j ‡ 12i ˆ
1
;
0
j ÿ 12i ˆ
0
:
1
sx , sy , sz are then represented by 2 2 matrices. It is convenient to take
out a factor »=2† and write s ˆ »=2†r ˆ »=2† x ; y ; z †. It is easy to
verify that the commutation relations are satis®ed using the Pauli
matrices:
x ˆ
0
1
1
;
0
y ˆ
0
i
ÿi
;
0
z ˆ
1
0
0
:
ÿ1
j ‡ 12i and j ÿ 12i are eigenvalues of z with eigenvalues ‡1 and ÿ1.
C.3
Addition of angular momenta
The total angular momentum of a spin 12 fermion is the sum of its orbital
and intrinsic angular momenta:
J ˆ L ‡ s:
It is easy to see that J satis®es commutation relations similar to (C.3), and
also ‰J; L2 Š ˆ 0, ‰J; s2 Š ˆ 0. It is therefore possible to ®nd states which are
simultaneous eigenstates of L2 , s2 , J2 and jz , speci®ed by quantum numbers l; s; j; jz †. These states have parity ÿ1†l .
For a given value of l and s ˆ 12 there are 2 2l ‡ 1† ˆ 4l ‡ 2 independent states, Ylm j 12i. We seek the linear combinations of these which
are the eigenstates of J2 and Jz . Since Jz ˆ Lz ‡ sz , the maximum value of
jz is l ‡ 12, corresponding to the state Yll j ‡ 12i. This must also be the
maximum value of j, and the state must also be an eigenstate of J2
corresponding to j ˆ l ‡ 12, jz ˆ l ‡ 12.
There are two independent states giving jz ˆ l ÿ 12, i.e. Yl;lÿ1 j ‡ 12i,
Yl;l j ÿ 12i. From these we must be able to construct the state correspond-
234
Appendix C
ing to j ˆ l ‡ 12, jz ˆ l ÿ 12. Another independent state can also be constructed; this clearly must correspond to j ˆ l ÿ 12, jz ˆ l ÿ 12.
The value j ˆ l ‡ 12 gives 2 l ‡ 12† ‡ 1 ˆ 2l ‡ 2 states; the value
j ˆ l ÿ 12 gives 2 l ÿ 12† ‡ 1 ˆ 2l states. Altogether, we have 4l ‡ 2† independent states, corresponding to the values j ˆ l ‡ 12, j ˆ l ÿ 12, and there
can be no more allowed values of j. We can think of the intrinsic spin s of
the particle as either aligned or anti-aligned with the orbital angular
momentum vector L, in so far as the uncertainty principle allows.
More generally, for two particles, or two systems, with angular
momenta J1 and J2 , we may form
J ˆ J1 ‡ J2 :
By an extension of the argument above, it can be shown that, for given
values of j1 and j2 , the allowed values of j are
j ˆ j1 ‡ j2 ; j1 ‡ j2 ÿ 1; . . . ; j j1 ÿ j2 j;
so that
j j1 ÿ j2 j 4 j 4 j1 ‡ j2 :
C.4
The deuteron
The total intrinsic spin S of two spin
1
2
fermions is
S ˆ s1 ‡ s2 ;
where from the rules above the quantum number S can take the values
S ˆ 1 and S ˆ 0. Explicitly, the three S ˆ 1 states jS, Sm i are found to be
j1; 1i ˆ j ‡ 12i1 j ‡ 12i2
p
j1; 0i ˆ j ‡ 12i1 j ÿ 12i2 ‡ j ÿ 12i1 j ‡ 12i2 †= 2
j1; ÿ1i ˆ j ÿ 12i1 j ÿ 12i2
C:6†
and the S ˆ 0 state is
p
j0; 0i ˆ j ‡ 12i1 j ÿ 12i2 ÿ j ÿ 12i1 j ‡ 12i2 †= 2:
(The factors
p
2 are for normalisation.)
C:7†
Unstable states and resonances
235
The deuteron is a neutron±proton bound pair having total spin
J ˆ L ‡ S with quantum number j ˆ 1 and total intrinsic spin S with
quantum number S ˆ 1. Neglecting a small l ˆ 2 wave component, its
spatial wave-function is an l ˆ 0 state. The wave-function of a deuteron
at rest is therefore approximately of the form
u r†j1; ms i;
where r is the distance between the two nuclei.
From (C.6), it will be seen that this wave-function is symmetric under
the interchange of proton and neutron. Thus such a state is not accessible
to two protons, or to two neutrons, since the wave-functions of two
identical fermions must be anti-symmetric under particle interchange
(}1.1). Although two nucleons with net intrinsic spin zero experience a
strong attraction, this attraction is not suf®cient to produce a bound state
and the deuteron is the only bound state of two nucleons.
Problems
C.1 Show that l ˆ 0 wave-functions ý r† (functions only of the radial coordinate r) are also eigenstates of Lx , Ly , Lz .
C.2 Explain why the single particle states speci®ed by l; s; j; jz † introduced in
}C.3 have parity ÿ1†l .
C.3(a) Show that the state j0; 0i given by equation (C.7) is an eigenstate of
Sx ˆ s1x ‡ s2x †, Sy and Sz and hence that it has total spin zero.
(b) Show that
Sz j1; 1i ˆ »j1; 1i
and
S2 j1; 1i ˆ 2 »2 j1; 1i:
Appendix D
Unstable states and resonances
In discussing unstable states, we have in mind a system like an excited
nucleus, or a þ-unstable nucleus. An unstable state of a system will decay,
and often there are several alternative modes of decay. For example, an
excited state of a nucleus can have several states of lower energy to which
236
Appendix D
it can decay by emitting a photon, and some þ-unstable nuclei can decay
by either þ‡ or þÿ -emission. Such distinct modes of decay are called
decay channels.
An unstable state has certain probabilities per unit time, called partial
decay rates, to decay into any of its channels. We shall denote these
probabilities by 1=i , where the i have the dimension of time and i labels
the ith decay channel. The total decay rate 1= is the sum of the partial
decay rates:
1 X1
ˆ
:
i
i
D:1†
We shall also ®nd it useful to de®ne partial widths ÿi and total widths
ÿ by ÿi ˆ »=i , ÿ ˆ »=. These have the dimensions of energy, and
clearly
ÿˆ
X
i
ÿi :
D:2†
The probability that the unstable state will decay to the ith channel is
the ratio of the partial decay rate into that channel to the total rate, i.e.
ÿi =ÿ.
For many of our applications it will be important that ÿ is a small
energy on the nuclear energy scale of MeV. For example, in ÿ-decay a
mean life 10ÿ14 s corresponds to ÿ 0:1 eV.
We have seen (}2.3) that a decay rate 1= implies that a state will
decay according to the exponential law
P t† ˆ P 0†eÿt= ;
where P t† is the probability of the state surviving at time t. Thus we can
identify the total decay rate with the inverse of the mean life.
D.1
Time development of a quantum system
We denote the wave-function of the unstable state by ý0 , and the states
into which it can decay by ý1 ; ý2 ; . . . ; ým ; . . . . (For example, the state ý0
might be that of a nucleus prior to -decay, and the states ým m > 0†
describe the residual nucleus and -particle in their ground states, and the
energy and direction of their relative motion.) We shall use periodic
boundary conditions, supposing our system enclosed in a large volume
Unstable states and resonances
237
V, so that all the states are discrete and may be normalised to unity. They
can always be chosen to be orthogonal to each other. We may therefore
take
Z
ým ýn dq ˆ mn ;
where dq d(all relevant coordinates).
ý0 is not an exact energy eigenstate: if it were, it would not decay.
Thus the state þ t† of the system, which is ý0 at t ˆ 0, develops an
admixture of the ®nal states. We can express þ t† as a superposition of
the states ým , and write
þ t† ˆ
1
X
mˆ0
am t†eÿiEm t= » ým :
D:3†
The phase factors, with
Z
Em ˆ Hmm ˆ
ým Hým dq;
where H is the Hamiltonian of the system, have been inserted for convenience. If all the states were exact eigenstates of H, the coef®cients am
would not depend on time. RHowever, we are interested in the case when
the matrix elements Hmn ˆ ým Hýn dq are in general non-vanishing for
m 6ˆ n. Inserting the expansion (D.3) in the SchroÈdinger equation
i»
@þ
ˆ Hþ
@t
gives
X
m
i »a_m eÿiEm t= » ým ‡ Em am eÿiEm t= » ým † ˆ
X
m
am eÿiEm t= » Hým :
Multiplying by ýn and integrating, the orthogonality relation picks out
the time dependence of an :
i »a_n ˆ
X
m6ˆn
(noting En ˆ Hnn †.
Hnm eÿi Em ÿEn †t= » am
D:4†
238
Appendix D
So far our equations are exact. The initial conditions at t ˆ 0 are
a0 0† ˆ 1, am 0† ˆ 0 for m 5 1.
We now work to ®rst order in the quantities Hnm , supposed small
when n 6ˆ m. Then for n 5 1 we have approximately
i »a_n ˆ Hn0 eÿi E0 ÿEn †t= » †a0 :
D:5†
The state ý0 is unstable. We make the ansatz that a0 t† ˆ eÿÿt=2 » so
that ja0 t†j2 ˆ eÿt= , and the probability of ®nding the system in the state
ý0 decays exponentially with time. The equations (D.5) can then be integrated to give
Z
i »an t† ˆ Hn0
t
0
eÿi‰ E0 ÿEn †ÿiÿ=2Št = » dt 0
0
(
)
0
»
eÿi‰ E0 ÿEn †ÿiÿ=2Št = » ÿ 1
ˆ Hn0
:
i
En ÿ E0 † ‡ iÿ=2
For times t 4 »=ÿ, eÿÿt=2 » ! 0 and for such times
an t† ˆ
Hn0
:
En ÿ E0 † ‡ iÿ=2
Thus the probability of decay to the state ýn is
jan t†j2 ˆ
2
jHn0 j2 P En ÿ E0 †;
ÿ
where
P En ÿ E0 † ˆ
ÿ
1
:
2 En ÿ E0 †2 ‡ ÿ2 =4
The function P E ÿ E0 † is shown graphically in Fig. D1. The factor ÿ=2
has been inserted so that
Z
1
ÿ1
P E ÿ E0 † dE ˆ 1:
An important aspect of our result which is exhibited in this ®gure is that
the energy of the ®nal state En is not identically equal to E0 , and indeed is
not absolutely determined. This feature is not to be interpreted as a
Unstable states and resonances
239
Fig. D1 The function P E ÿ E0 † ˆ ÿ=2†‰ E ÿ E0 †2 ‡ ÿ2 =4Šÿ1 .
violation of energy conservation, but as a consequence of the fact that the
state ý0 does not have a de®nite energy. The instability of the state
implies that it has a small spread of energy of width ÿ about its mean
R
energy E0 ˆ ý0 Hý0 dq. The function P E ÿ E0 † can be regarded as the
probability distribution in energy of the state ý0 .
It is interesting to remark that we can now interpret the relationship
ÿ ˆ »
as a relationship between uncertainty in energy and lifetime, somewhat
similar to the Heisenberg uncertainty relation between momentum and
position.
To obtain the probability of decay to a channel i, we must sum over
all the states n in i. For example, consider the -decay of 238U nucleus at
rest:
238
92 U
4
! 234
90 Th ‡ 2 He:
240
Appendix D
In this case, i is the -decay channel. All the nuclei involved have zero
spin, so that the states n in channel i are completely speci®ed by their
energy, and the direction of emission of the -particle. Since there are no
spin orientations to be considered, the matrix element Hn0 will not depend
on the direction of emission, and the probability of decay to channel i is
X
2
jan t†j ˆ
ÿ
n in i
2
Z
jHn0 j2 P E ÿ E0 †ni E† dE;
where ni E† is the density of states in channel i at energy E (Appendix B).
For ÿ small, the integral comes almost entirely from around the peak
in P E ÿ E0 † at E0 . Assuming that ni E† and jHn0 j2 vary slowly with E
over the width of the peak, we may evaluate them at E0 and treat them as
constant in the integration to give
X
n in i
jan t†j2 ˆ
2
jHn0 j2 ni E0 †:
ÿ
Since the probability of decay to channel i is simply ÿi =ÿ, it follows
that the partial decay rate, when no spins are involved, is
1 ÿi 2
jHn0 j2 ni E0 †:
ˆ ˆ
»
i
»
D:6†
This result is known as Fermi's golden rule.
In the more general situation when the decay products have spin, and
the initial unstable state has spin j, we will include in the channel i all the
spin states of the ®nal particles, and consider the case when the spin of the
unstable state is not polarised in any particular direction. We must then
average over all 2j ‡ 1† initial spin states. After averaging, the result does
not depend on direction and the formula (D.6) becomes
1 ÿi 2 ni E0 † X
jHn0 j2 ;
ˆ ˆ
»
i
» 2j ‡ 1
D:7†
where the sum is over all initial spin states and ®nal spin states, and ni E†
is the density of states neglecting spin.
Unstable states and resonances
D.2
241
The formation of excited states in scattering: resonances and
the Breit±Wigner formula
We now consider a channel i which consists of two particles. We take as
an example a neutron interacting with a nucleus I at an energy close to an
energy at which the two can combine to form the unstable excited state
X . If created, X will then decay into one of its decay channels, say
channel f . The overall process can be represented by
n ‡ I ! X ! channel f †:
Such scattering processes which proceed through an intermediate
unstable state have important characteristics we wish to discuss. We consider a situation where initially the amplitude a0 of the unstable state is
zero, i.e. a0 0† ˆ 0, and the system is in an initial state, ý1 say, which
belongs to channel i, so that a1 0† ˆ 1. The amplitude a0 develops in time
according to the exact equation (D.4) with n ˆ 0:
i »a_0 ˆ
X
m6ˆ0
H0m eÿi Em ÿE0 †t= » am :
Again working to ®rst order in the small quantities H0m we have
i »a_0 ˆ ÿi ÿ=2†a0 ‡ H01 eÿi E1 ÿE0 †t= » :
The term involving ÿ which we have introduced gives the decay of the
unstable state in accordance with our ansatz and takes account, in a
phenomenological way, of the small terms in the exact equation that
have otherwise been neglected.
We can write this equation as
i»
d
a eÿt=2 » † ˆ H01 eÿi E1 ÿE0 ‡iÿ=2†t= » ;
dt 0
so that
i »a0 e
ÿt=2 »
Z
ˆ
t
0
0
H01 eÿi E1 ÿE0 ‡iÿ=2†t = » dt 0 :
For times t long compared with »=ÿ we obtain
242
Appendix D
a0 t† ˆ
H01 eÿi E1 ÿE0 †t= »
;
E1 ÿ E0 ‡ iÿ=2
and the probability of ®nding the state ý0 is
ja0 t†j2 ˆ
jH01 j2
:
E1 ÿ E0 †2 ‡ ÿ2 =4
The decay rate into the channel f is therefore
ja0 t†j2
ÿf
1
jH01 j2
:
ˆ
2
2
f
E1 ÿ E0 † ‡ ÿ =4 »
D:8†
Suppose, for the moment, that the initial particles and the excited
state have spin zero. Then it is useful to de®ne
ÿi E† ˆ 2jH10 j2 ni E†
D:9†
which is a generalisation of (D.6), and, since jH10 j2 ˆ jH01 j2 , we can rewrite the decay rate into channel f as
ÿi E1 †ÿf
1
1
:
2 » ni E1 † E1 ÿ E0 †2 ‡ ÿ2 =4
If the relative motion of the interacting particles is given by the wave1
function V ÿ2 eikr (in the centre-of-mass coordinate system; see Appendix
A), the ¯ux of particles is given by (particle density) velocity = V ÿ1 v.
The cross-section 1 ! f † for scattering into channel f is de®ned by
flux of 1† 1 ! f † ˆ decay rate into channel f :
Hence
1!f†ˆ
ÿi ÿf
V 1
1
:
v 2 » ni E1 † E1 ÿ E0 †2 ‡ ÿ2 =4
The density of states in the ith channel is given by (Appendix B)
ni E† dE ˆ
V
dk
dE;
4k2
3
dE
2†
Unstable states and resonances
243
and dE=dk ˆ »2 k=m ˆ »v. (m is the reduced mass of the particles.)
Hence, substituting, we obtain
1!f†ˆ
ÿi ÿf
:
k21 E1 ÿ E0 †2 ‡ ÿ2 =4
D:10†
This result is the Breit±Wigner formula for the special case when the
incoming particles and the compound nucleus have zero spin. For small
ÿ, the cross-section peaks sharply at E1 ˆ E0 . The phenomenon is known
as resonance scattering and is common in nuclear physics; experimental
resonance peaks can often be well ®tted by an expression of this form.
The formula for the general spin case is more complicated. Suppose
the initial spins of the particles are s1 and s2 . For example, for neutrons
interacting with 234U the spin of the neutron is s1 ˆ 12 and the spin of 235U
is s2 ˆ 72. If, as in a nuclear reactor, the neutrons and the uranium nuclei
are not polarised, then we have to average the cross-section over the
2s1 ‡ 1† 2s2 ‡ 1† initial spin states. Consider also the formation of
an excited state of 236U with spin j. Any of its 2j ‡ 1† sub-states can
be formed, and they all contribute to the production of the ®nal state.
Equation (D.8) which gives the decay rate into channel f (a ®ssion channel, for example) has to be modi®ed to:
Decay rate into channel f
ˆ
X
ÿf
1
1
jH j2 :
2
2
2s1 ‡ 1† 2s2 ‡ 1† » E1 ÿ E0 † ‡ ÿ =4 spins 01
This, using (D.7) and (D.9), yields the general Breit±Wigner formula:
1!f†ˆ
ÿi ÿf
2j ‡ 1†
:
2 2s ‡ 1† 2s ‡ 1†
k1 1
E1 ÿ E0 †2 ‡ ÿ2 =4
2
D:11†
The total cross-section is obtained by summing over all channels f :
tot ˆ
2j ‡ 1†
ÿi ÿ
:
k21 2s1 ‡ 1† 2s2 ‡ 1† E1 ÿ E0 †2 ‡ ÿ2 =4
D:12†
The mechanism of formation of the unstable states, and their subsequent
decay, is discussed in more detail in Chapter 7.
This expression is a good approximation when one unstable state
dominates the cross-section, but scattering which proceeds by other
244
Appendix D
mechanisms than the formation of an unstable state is not included in our
discussion. For example, direct reactions, mentioned in Chapter 7, are not
included.
Problems
D.1 The Breit±Wigner formula of }D.2 was derived for a particle incident
upon a nucleus. It has to be modi®ed if, as in the case of ± scattering,
the `particle' and the nucleus are identical. Show that for ± resonant
scattering through the formation of 84 Be
ˆ
2
ÿ2
:
k21 E1 ÿ E0 †2 ‡ ÿ2 =4
(See Problem B.1.)
Further reading
Texts at a somewhat more advanced level than this one include:
Jelley, N. A. (1990), Fundamentals of Nuclear Physics,
Cambridge: Cambridge University Press.
Krane, K. S. (1987), Introductory Nuclear Physics, New York:
Wiley.
Wong, S. S. M. (1998), Introductory Nuclear Physics (2nd edn.),
New York: Wiley.
The student may also ®nd interesting:
Cameron, I. R. (1982), Nuclear Fission Reactors, New York:
Plenum.
Clayton, D. D. (1983), Principles of Stellar Evolution and
Nucleosynthesis, Chicago: University of Chicago Press.
Cottingham, W. N. and Greenwood, D. A. (1998), An
Introduction to the Standard Model of Particle Physics,
Cambridge: Cambridge University Press.
Phillips, A. C. (1994), The Physics of Stars, Chichester: Wiley.
Pochin, E. (1983), Nuclear Radiation: Risks and Bene®ts, Oxford:
Clarendon Press.
245
Answers to problems
(Unless otherwise stated, the mass of a nucleus of mass number A is
approximated as A amu.)
Chapter 1
1.1 Ratio = Gm2e 4"0 =e2 † ˆ 2:4 10ÿ43 .
1.2(b) (i) +1, (ii) ÿ1, (iii) ÿ1.
1.3(a) Wavelength
ˆ 2c=! ˆ 2 »c†= »!† ˆ 2 197 MeV fm†= 1 MeV† ˆ 1240 fm.
Chapter 2
2.1 Group velocity = d!=dk ˆ c2 k=! ˆ c2 »k= »! ˆ c2 p=E.
For a particle of velocity v, E ˆ ÿmc2 and p ˆ ÿmv.
Hence group velocity = particle velocity.
2.2 From equation (2.8) the electrostatic energy is of order of magnitude
e2 =4"0 a0 .
From equation (2.13) and after, the weak interaction energy is of order
of magnitude "ÿ1
»=MZ c†2 e2 =a30 .
0
Ratio = 4 »=a0 MZ c†2 10ÿ15 .
2.3 By momentum conservation the momenta of the two photons must be
equal in magnitude (and opposite in direction). They will therefore have
equal energy.
2.4 There is a frame of reference (the centre-of-mass frame) in which the
total momentum of the e‡ eÿ pair is zero. The photon would therefore
246
Chapter 3
247
have zero momentum and hence zero energy: energy conservation would
be violated.
2.5 Since E ˆ 1 MeV is much less than the rest energy of the muon
we may use non-relativistic mechanics, and the velocity
p
v ˆ c 2E=m c2 † ˆ 4:1 107 msÿ1 . In time the muon travels
v ˆ 90 m.
2.6(a) Not allowed. Such a process need not violate the conservation laws of
energy, momentum, angular momentum or electric charge, but it would
violate the conservation laws of electron number and muon number.
Although searched for, this decay has never been seen. (b) and (c) can
occur.
2.8 Consider a point charge e at position R; then
E r† ˆ
e rÿR
:
4"0 jr ÿ Rj3
Under re¯ection in the origin, r ! r 0 ˆ ÿr, R ! R 0 ˆ ÿR and
E 0 r 0† ˆ
e r0 ÿ R0
ˆ ÿE r†:
4"0 jr 0 ÿ R 0 j3
The magnetic ®eld due to a current I in a loop is
Z
I dR r ÿ R†
B r† ˆ 0
:
4
jr ÿ Rj3
Under re¯ection the vector product does not change sign. Hence
B 0 r 0 † ˆ ‡B r†; B r† is an axial vector ®eld.
2.9 The reduced masses are
m md = m ‡ md † ˆ 100:025 MeV for d± system,
m mt = m ‡ mt † ˆ 101:829 MeV for t± system.
The difference in binding energies is
1
m†c2 e2 =4"0 »c†2 ˆ 48 eV:
2
Chapter 3
3.1 The nucleon magnetic dipole moments are vectors aligned with the
nucleon spin. In the deuteron the spins are parallel and the moments
add to give a net magnitude p ‡ n ˆ 2:792 84 ÿ 1:913 04†e »=2mp ˆ
248
Answers to problems
0:8798e »=2mp (from equation (3.2)).
The measured magnitude is
d ˆ 0:8574e »=2mp ˆ 1 ÿ 0:026† p ‡ n †:
The discrepancy (0.026) could be due to a contribution to the magnetic
moment from the orbital motion of the nucleons, associated with the
small d-wave component of the deuteron wave-function. (Appendix C,
}C.4.)
3.2(a) The magnetic ®eld at distance r from dipole (1) is
r r r
3 r1 r†r
1
0
1
ˆ
:
Bˆÿ 0 r
ÿ
‡
4
4
r5
r3
r3
The energy of dipole (2) at r in this ®eld is
2
ÿr2 B ˆ ÿ 0 3 þT :
4 r
3.3 Subtracting 1 MeV from the rest energies of the charged particles gives
(udd) 940, (uud) 937; 3 MeV for the extra d quark;
(dds) 1196, (uds) 1192, (uus) 1188; 4 MeV for the extra d quark;
(ds) 498, (us) 493; 5 MeV for the extra d quark.
Interchanging a d quark for a u quark always increases the rest energy,
in this sample by an average of 4 MeV.
3.4(a) Not allowed. Does not conserve electric charge.
(b) Not allowed. Does not conserve baryon number or electron number.
(c) Not allowed. Does not conserve baryon number.
(d) Allowed.
3.5 ÿ ! Kÿ ‡ 0
0 ! 0 ‡ ÿ
0 ! p ‡ eÿ ‡ e
Kÿ ! ÿ ‡ 0
0 ! ÿ ‡ ÿ
ÿ ! ÿ ‡ ÿ ! eÿ ‡ e ‡ Strong. Does not require the weak or the electromagnetic interaction.
Electromagnetic.
Involves an anti-neutrino, therefore weak.
An s quark changes to a d quark, therefore weak.
Electromagnetic (Fig. 3.6).
Weak (cf. Fig. 3.5).
Weak (Fig. 2.2).
Chapter 4
4.2(a) Since q ˆ kf ÿ ki , where ki and kf are the initial and ®nal wave vectors,
q2 ˆ k2f ‡ k2i ÿ 2kf ki cos :
Chapter 4
249
Neglecting the electron mass, ki ˆ pi = » ˆ E= »c, and in scattering from
a ®xed target there is no energy loss. Hence kf ˆ ki ˆ E= »c and
q2 ˆ 2E 2 1 ÿ cos †= »2 c2 ;
q ˆ 2E= »c† sin =2†:
4.3 By the uncertainty principle, the mean magnitude of the lepton momen1
tum p »=a. But p ˆ mv, so that v=c† »=amc ˆ e2 =4"0 »c ˆ 137
.
ÿ19
The characteristic time t ˆ a=v ˆ 137a=c ˆ 1:2 10
s, and
=t† 2 1013 .
4.4(b) This follows from perturbation theory in quantum mechanics. Since the
integral is over nuclear dimensions r 4 R, it is reasonable to approxi1
3
mate ý r† by ý 0† ˆ ÿ2 Z=a†2 , which with Problem 4.4(a) gives the
result.
4.5 The total binding energy of two -particles is 56.60 MeV, 0.1 MeV
greater than the binding energy of 84 Be. 84 Be decays to two -particles
and to conserve energy the 0.1 MeV of nuclear energy is converted into
the kinetic energy of their motion. 126 C is more strongly bound than three
-particles by 7.26 MeV. The binding energy of 63 Li is 31.99 MeV,
1.47 MeV greater than the total binding energy of 21 H and 42 He.
Energy is conserved overall, and the nuclear energy released is taken
by the ÿ-ray and the kinetic energy of the 63 Li.
4.6 Treating A as a continuous variable, the maximum is where
d B=A†=dA ˆ 0, i.e. at
A
b
ˆ Z ˆ ˆ 25:7
2
d
The nearest integer is Z ˆ 26, which gives the maximum of B=A†.
4.7 For A ˆ 100, the formula gives Z ˆ 43. 100
43 Tc is an odd±odd nucleus and
100
Mo
and
Ru
are
stable.
For A ˆ 200, the formula
unstable. Both 100
42
44
gives Z ˆ 80, and 200
Hg
is
stable.
80
4.8 Suppose the sample contains N 14C nuclei. Then the mean number of
decays per second is N= ˆ 15:3=60 sÿ1 , and hence N ˆ 6:7 1010 . The
atomic mass of natural carbon is 12.01 amu = 2 10ÿ23 g. Therefore
1 g of carbon contains 5 1022 atoms, and the proportion of 14C in the
sample is 1:3 10ÿ12 .
Assuming (i) that the proportion of 14C in the atmosphere remains
constant, (ii) that the hut had been built from new timber, and (iii) that
the carbon in the timber had all been bound in at the time of its growth,
the average number of decays per minute would be reduced by a factor
eÿt= , where is the mean life of 14C and t the age of the specimen. Thus
the expected rate would be 9.4 decays per minute. Conversely, an average of 9.4 decays per minute would suggest an age of 4000 years.
250
Answers to problems
In practice, carbon dating is far more complicated than this problem
suggests. In the past there have been small variations in the proportion
of 14C in the atmosphere, as appears when radiocarbon dating can be
calibrated against dating obtained by other methods, such as dendrochronology (counting tree rings). These variations can lead to ambiguities in the inversion of the calibration curve.
82
4.9 The
number
of
atoms
of
Se
in
the
sample
is
ÿ24
N ˆ 0:97 14 g†= 82 1:66 10
g† 1:00 1023 . If is the mean
life, in a time t there will be Nt= decays, giving
ˆ Nt=(number of events)
ˆ 1023 7960 hr/(35/0.062)
ˆ 1:6 1020 yr.
(The condition t is well satis®ed!)
4.10 In the absence of neutrinos to share the energy released in the decay, the
sum of the energies of the two electrons emitted would be sharply peaked
at the decay energy. (The recoil energy of the nucleus would be small.)
4.11 As a rough rule, for A odd there is only one þ-stable nucleus and for A
even, two. Up to and including A ˆ 209 there are 105 odd-A nuclei and
104 even-A nuclei, and so about 310 þ-stable nuclei. All these have
Z 4 83 which implies an average of about 3.7 stable isotopes per element.
Chapter 5
5.2(a) From equations (5.4) and (5.5), for neutrons with kinetic energy E,
3
N E† ˆ N E=EnF †2 .
1
3
The density of states is n E† ˆ dN =dE ˆ 32 NE 2 EnF †ÿ2 .
Hence the total neutron kinetic energy is
Z EnF
3
En E†dE ˆ NEnF :
5
0
A similar result holds for the protons, noting that proton kinetic energies
are given by Ep ÿ U†.
2
(b) Again consider the neutrons; since EnF is proportional to N 3 we can write
2
2
EnF ˆ E0F N=N0 †3 ˆ E0F 1 ‡ N=N0 †3 :
The result follows on expansion.
2
Similarly EpF ÿ U ˆ E0F 1 ÿ N=N0 †3 :
Chapter 5
(c)
251
5
F
3
5 NEn
ˆ 35 N0 E0F 1 ‡ N=N0 †3
ÿ
2 !
5 N
5 N
F
3
:
‡
5 N0 E0 1 ‡
3 N0
9 N0
Adding the proton kinetic energy, one obtains the total kinetic energy
N ÿ Z†2
:
A
ÿ
!
ÿ
!ÿ1 Z
2
3
2
R
Z
ÿ
1†e
4R
3
r
2
5.3 U c ˆ
dr
4r
ÿ
4"0 R
3
2 2R2
0
3 F
5 E0 A
ˆ
‡ 13 E0F
6 Z ÿ 1†e2
21:5 MeV
5 4"0 R
for208
82 Pb:
5.4 Since Z ˆ N, the energy due to the strong nucleon±nucleon interaction
should be the same for a neutron as for a proton in a similar state; thus
U should be given by the Coulomb contribution, U c of Problem 5.3(a).
This yields U ˆ 8:7 MeV. The separation energy is the binding energy at
it follows that
the Fermi level. Assuming EpF EnF ‡ U,
Sp ˆ 15:6 ÿ 8:7† MeV.
5.5 In the simple shell model, 31
15 P has an odd proton; Table 5.1 suggests this
‡
is in the 2s12 shell, giving nuclear spin and parity j P ˆ 12 .
67
30 Zn
115
49 In
ÿ
has an odd neutron. Suggested shell f 52 ; j P ˆ 52 .
‡
has an odd proton. Suggested shell 1g92 ; j P ˆ 92 .
These suggestions are all in agreement with experiment.
5.6 The spins and parities of all but 26
13 Al are in accord with pairing and shell
®lling as in Table 5.1. 26
13 Al is odd±odd; the model suggests the odd
‡
neutron and proton both to be in 52 states. Such a con®guration
26
would have the measured parity of 13 Al; and the measured spin of 5
suggests that the spins are paired parallel.
For the magnetic moments, equation (5.26) gives:
43
20 Ca, odd neutron,
93
41 Nb, odd proton,
137
56 Ba, odd neutron,
197
79 Au, odd proton,
l
l
l
l
ˆ 3;
ˆ 4;
ˆ 2;
ˆ 2;
j
j
j
j
ˆ l ‡ 12 ;
ˆ l ‡ 12 ;
ˆ l ÿ 12 ;
ˆ l ÿ 12 ;
ˆ ÿ1:92 N
ˆ 6:80 N
ˆ 1:15 N
ˆ 0:12 N .
26
13 Al. The fact that the two angular momenta appear to be aligned
suggests that we can simply add the Schmidt values to obtain the estimate ˆ 2:9 N .
5.7 For a proton, j ˆ 12 ; ˆ 2:80 N .
7
For 43
20 Ca, j ˆ 2 ; ˆ ÿ1:32 N .
252
Answers to problems
Taking the nuclear magneton N ˆ 3:15 10ÿ14 MeV Tÿ1 and using
equation (5.21) gives
ˆ !=2 ˆ 43 MHz
ˆ !=2 ˆ 2:9 MHz
for protons,
for 43
20 Ca.
5.8 The volume of the ellipsoid is 4=3†a2 b and hence the charge density is
3Ze=4a2 b.
ZZZ
3Z
2z2 ÿ x2 ÿ y2 † dx dy dz;
Qzz ˆ
4a2 b
where the integral is through the ellipsoid.
Make a change of scale: x ˆ ax 0 ; y ˆ ay 0 ; z ˆ bz 0 ; then
ZZZ
0
0
0
3Z
Qzz ˆ
2b2 z 2 ÿ a2 x 2 ÿ a2 y 2 † dx 0 dy 0 dz 0 ;
4
where the integral is now through the unit sphere. Also
Z
ZZZ
0
0
1 1 02
4
; etc:;
x 2 dx 0 dy 0 dz 0 ˆ
r 4r 2 dr 0 ˆ
3 0
15
giving
Qzz ˆ
2Z 2
b ÿ a2 †:
5
Taking the density of nuclear matter to be 0 ˆ 0:17 nucleons fmÿ3 and
4 2
a b0 ˆ A;
3
these equations lead to b ˆ 7:7 fm, a ˆ 5:6 fm.
Chapter 6
6.1 Q ˆ 0:094 MeV,
1
rs 2 43 1:1 fm = 3.5 fm,
rc ˆ 4e2 =4"0 Q ˆ 61 fm:
rs =rc ˆ 0:057 and Fig. 6.3 or equation (6.16) gives G = 0.70.
The reduced mass is m =2. Hence G ˆ 13.
Taking 0 7 10ÿ23 s, as in other -decays, leads to the estimate
ˆ 3 10ÿ17 s, though this excellent agreement is fortuitous.
6.2(a) To apply equations (6.2) and (6.15) to a positron, 2Zd ! Zd and m
becomes the positron mass.
1
Thus rc ˆ 112 fm. Values of rs between rs ˆ 0 and rs ˆ 1:1A3 fm =
Chapter 7
253
6.37 fm are reasonable, giving G=1, 0.70; G=1.81, 1.27, and a suppression factor eÿG in the range 0.16 to 0.28.
(b) Q ˆ 1:8 Mev, rc ˆ 123 fm, rs 8:1 fm, G = 0.68, G ˆ 154,
which with 0 ˆ 7 10ÿ23 s gives a partial mean life for -decay
1037 years.
(Age of Solar System 109 years.)
6.3 In -decay in material, the kinetic energy is largely converted into heat
and N atoms of 238Pu would on average produce N 5:49 MeV†= of
power.
For 1 kW ˆ 6:24 1015 MeV sÿ1 we need N ˆ 4:6 1024 , or 1.8 kg of
238
Pu.
The decay rate of the by-product 234U is so low that the heat from its
decay is negligible.
For the remaining mass of plutonium to be 1.8 kg after 50 years requires
2.7 kg initially.
6.4 Suppose that when the sample of rock was formed, say T years ago, it
contained no lead but N1 atoms of 238U (mean life 1 ) and N2 atoms of
235
U (mean life 2 ). Then it would now contain N1 eÿT=1 atoms of 238U
and, since each decayed uranium atom becomes a lead atom,
N1 1 ÿ eÿT=1 † atoms of 206Pb. Setting 1 ÿ eÿT =1 †=eÿT=1 ˆ 0:0797 suggests T ˆ 497 106 years. Similarly for 235U and 207Pb,
1 ÿ eÿT=2 †=eÿT=2 ˆ 0:675 suggests T ˆ 531 106 years. (The discrepancy could be due to the effect of water on the rock, for example.)
6.5 Neglecting the excitation energy, the kinetic energy of the fragments can
be estimated using equation (6.18), which gives B ˆ 178 MeV. Each
fragment would then have velocity 12 106 m sÿ1 . In the frame in which
the fragment is at rest, a 2 MeV neutron has velocity 20 106 m sÿ1 . In
the laboratory frame, the distribution of emitted neutrons is peaked in
the direction of the moving fragment.
Chapter 7
7.3(a† Ei ‡ E0 ˆ Ef ‡ (excitation energy) + (17O* recoil energy).
Hence the recoil energy is 0.26 MeV and the recoil velocity is
v=c ˆ 5:7 10ÿ3 (approximating the mass of 17O* by 17 amu).
(b) If the photon has energy Eÿ it has momentum Eÿ =c, and to conserve
momentum this must be the recoil momentum of the 17O. Hence the 17O
recoil energy ER is ER ˆ Eÿ =c†2 = 34 amu).
To conserve energy, Eÿ ‡ ER ˆ 0:87 MeV.
We could solve these equations for Eÿ , but clearly ER is small, and to
two signi®cant ®gures it is suf®cient to take Eÿ ˆ 0:87 MeV in the ®rst
equation to give ER ˆ 24 eV.
254
Answers to problems
(c) The photon energy will be a maximum if it is emitted parallel to the
motion of the 17O*. By a Lorentz transformation to the laboratory
frame
1
Eÿlab ˆ 1 ‡ v=c†Eÿ = 1 ÿ v2 =c2 †2
and hence Eÿlab ÿ Eÿ ˆ 5 keV.
Similarly the photon energy will be a minimum if it is emitted antiparallel, in which case Eÿlab ÿ Eÿ ˆ ÿ5 keV.
7.4
11
6C
is less bound than 115 B by 2.762 MeV. The difference of Coulomb
energies of uniformly charged spheres of net charge 6e and 5e and radius
1
R ˆ 1:1 113 fm is
3 e2
62 ÿ 52 † ˆ 4 MeV:
5 4"0 R
This is a 50% over-estimate of the observed energy difference, and we
would need to take RC ˆ 1:45 R to obtain agreement. The approximation of a uniform charge distribution is inadequate for precise calculations, especially for light nuclei. In reality some charge is displaced to
larger distances (see Fig. 4.3) thereby reducing the energy. Calculations
using the more realistic distributions are in better accord with the data.
7.5 The decay by neutron emission with a kinetic energy release of 0.41 MeV
need involve only the strong interaction. There is no Coulomb barrier,
and only a small angular momentum barrier: to conserve angular
momentum and parity the angular momentum of the 16O±n pair must
be l ˆ 1.
The mean life is still quite long on the nuclear time scale of 10ÿ22 s.
ÿ ˆ »c=c ˆ 0:04 MeV.
ÿ
ÿ
7.6 The nuclear transition is 12 ! 32 , so the photon will have positive parity
and angular momentum quantum number 2 5 j 5 1. The most likely
transition is with j ˆ 1, which would be magnetic dipole. The photon
energy is about 2.13 MeV. From Fig. 7.6, a rough estimate of the mean
2
2
life is 10ÿ17 100=A†3 20A3 s 4 10ÿ15 s. (The experimental
mean life is 5:2 10ÿ15 s.) An electric quadrupole transition with j ˆ 2
is also possible, but Fig. 7.6 suggests its partial decay rate to be much
slower than the magnetic dipole rate.
7.7 The lowest six energy levels (comprising 26 states) all have positive
parity. 105 B has three protons in the p32 shell and three neutrons in the
p32 shell. There are many combinations of the single nucleon p-states and
they all have positive parity, ÿ1†6 . The lowest observed states can be
considered to be constructed from these.
The 1.74 MeV level can decay to the 0.72 MeV level by a magnetic
Chapter 8
255
dipole 0‡ ! 1‡ † transition with Eÿ 1:02 MeV. This level can in turn
decay to ground by an electric quadrupole 1‡ ! 3‡ † transition with
Eÿ 0:72 MeV. Neglecting internal conversion the ratio of photons
emitted is clearly one-to-one.
The 1.74 MeV level can also decay directly to ground with
Eÿ 1:74 MeV but this 0‡ ! 3‡ † transition is magnetic octapole and
very slow. Using Fig. 7.6 the number of photons emitted with energies
1.02 MeV, 0.72 MeV and 1.74 MeV should be in proportion 1:1:10ÿ8.
Chapter 8
8.2 A neutron with kinetic energy 0.1 eV has v=c ˆ 1:46 10ÿ5 giving
ˆ 2670 b, l 1:56 cm. The probability of a neutron penetrating a
distance x into the gas without interaction is eÿx=l . For this probability
to be 0.1, we require x ˆ 3:6 cm. The active region of the detector
should be at least of this thickness.
8.3(a) ˆ »=ÿ ˆ »c=ÿc ˆ 1:3 10ÿ21 s:
(b) In this example the elastic width equals the total width to a good approximation, since there is not enough energy to induce other nuclear reactions. The spin of the neutron is s1 ˆ 12 and the spin of 42 H is s2 ˆ 0.
Hence the statistical factor in the Breit±Wigner formula is
2j ‡ 1†= 2s1 ‡ 1† 2s2 ‡ 1† ˆ 2, and the cross-section at energy E is
E† ˆ
2
ÿ2
2
k E ÿ E0 †2 ‡ ÿ2 =4
equation (D.11)†:
At energy E ˆ E0 , ˆ 8=k2 ˆ 4 »2 =mE0 , where m is the reduced
mass. Hence 3:2 b.
8.4 The coef®cient of the 1=v† term is large if the incident neutron can easily
induce a nuclear reaction (as in the case of 235U ®ssion), or if there is an
excited state close to zero incident neutron energy. Neither of these
conditions is apparently satis®ed in the case of 238U. However, one
would expect to see a small 1=v† contribution at even lower neutron
energies due for example to residual radiative capture.
8.5 The total width ÿ is given approximately by ÿ ˆ ÿÿ ‡ ÿn , and the relative probability of neutron radiative capture is
ÿÿ =ÿ ˆ 1= 1 ‡ ÿn =ÿÿ †:
In this application of the Breit±Wigner formula the neutron spin s1 ˆ 12
and the spin of the even±even nucleus 238U is s2 ˆ 0. Also j ˆ 12. Hence at
resonance
256
Answers to problems
ÿ ˆ
4 ÿn ÿÿ
;
k2 ÿ n ‡ ÿ ÿ † 2
where k2 ˆ 2mE0 = »2 .
From Fig. 8.5, E0 ˆ 6:7 eV and ÿ ˆ 2 104 b. Hence
ÿn ÿÿ
ÿn ‡ ÿÿ †
2
ˆ
ÿn =ÿÿ
1 ‡ ÿn =ÿÿ †2
ˆ 0:052:
Since we are told that ÿn =ÿÿ is small, we take the solution
ÿn =ÿÿ ˆ 0:058. Hence capture is 95% probable.
Chapter 9
9.1 The molecules CH4 ‡ 2O2 have a mass 80 amu and release 9 eV in
chemical reaction, i.e. 0.11 eV per amu. 235U has a mass 235 amu and
releases about 200 106 eV on ®ssion (Table 9.1 and discussion), giving
0:85 106 eV per amu.
Ratio 8 106 .
9.3 Sn ˆ B N; Z† ÿ B N ÿ 1; Z†. The quoted difference comes from the
pairing energy terms. All other terms in the mass formula give contributions to Sn which for a heavy nucleus vary only slowly with A.
9.4 The probability of the neutron inducing ®ssion at the nth collision is
p 1 ÿ p†nÿ1 .
By de®nition the mean number of collisions is
n ˆ
1
X
np 1 ÿ p†nÿ1 ˆ ÿp
1
1
d X
1 ÿ p†n :
dp 1
Summing the geometric series
n ˆ ÿp
d
1
‰1 ÿ 1 ÿ p†Šÿ1 ˆ :
dp
p
9.5(a) If v is the neutron velocity in the laboratory frame, its velocity in the
centre-of-mass frame is
vÿ
mn v
Mv
ˆ
:
M ‡ mn M ‡ mn
In the centre-of-mass frame it loses no energy on scattering, but suppose
it is de¯ected through an angle . In the laboratory frame it will then
have a component of velocity v mn ‡ M cos †= M ‡ mn † in the original
direction and a perpendicular component Mv sin = M ‡ mn †. The result
follows on averaging over all angles .
Chapter 9
257
(b) On average, after N collisions a neutron with initial energy E0 will have
energy EN ˆ N E0 , where
ˆ
M 2 ‡ m2n
ˆ 0:86:
M ‡ mn †2
For E0 ˆ 2 MeV, EN ˆ 0:1 eV, the number of collisions required is
N 110.
The mean time between collisions for a neutron of energy E is
p
t ˆ 1=v ˆ 1= 2E=mn †, and it loses energy E ˆ 1 ÿ †E.
Approximating the mean rate of change of energy by
1
3
dE
E
ÿ
ˆ ÿ 1 ÿ † 2=mn †2 E 2
dt
t
gives the time to `cool' to EN ˆ 0:1 eV:
s
ÿ
!12
Z
1
mn c2 E0 dE
1
2mn c2
ˆ 8 10ÿ5 s:
time ˆ
3 1 ÿ †c
1 ÿ †c EN
2
EN E 2
9.6 From the text l ˆ 3 cm, l=v ˆ 1:5 10ÿ9 s, ˆ 2:5 and tp ˆ 9 10ÿ9 s.
Substituting the form r; t† ˆ f r†et into the equation yields
f r† ˆ
ÿ 1†
D d2
f r† ‡
rf r††;
tp
r dr2
which has solutions of the form f r† ˆ 1=r† sin kr† provided
ˆ
ÿ 1†
ÿ Dk2 :
tp
To satisfy the boundary condition,
k R ‡ 0:71l† ˆ n;
n ˆ 1; 2; . . .
To avoid an exponential increase in density 4 0 for all n and therefore
R ‡ 0:71l†2 4
i.e.
r
R4 tp
D2 ;
ÿ 1†
tp v
ÿ 0:71 l;
3 ÿ 1†l
and the critical radius in this approximation is 8.8 cm.
9.7(a) Suppose that the probability of a neutron induced ®ssion to result in a
fragment which produces a delayed neutron is d , and that the number
of such fragments at any time is N t†, then
258
Answers to problems
dN
N t† qd
ˆÿ
‡
n t†:
dt
þ
tp
This equation has the solution
N t† ˆ
qd
tp
Z
t
ÿ1
0
eÿ tÿt †=þ n t 0 † dt 0 :
Including delayed neutrons in equation (9.1) gives
q ÿ 1†
dn
N t†
n t† ‡
;
ˆ
dt
þ
tp
which is the equation quoted.
(b) ˆ
q ÿ 1†
d q
‡
tp
tp 1 ‡ þ †
or
þ †2 ‡ þ †‰1 ÿ q ÿ 1†þ =tp Š ÿ þ =tp †‰ ‡ d †q ÿ 1Š ˆ 0:
(c) Clearly ˆ q ÿ 1†=tp ˆ 1 s when d ˆ 0.
(d) Substituting the given values in the quadratic equation gives
þ †2 ‡ 781 þ † ÿ 10 ˆ 0.
The positive solution for þ † corresponds to an exponentially increasing n t†, with time scale 1= ˆ 13 min.
9.8(a) Take the energy release per ®ssion to be 178 ‡ 15† MeV (Table 9.1).
The number of ®ssions in time dt 0 is then Pdt 0 = 193 MeV†. For steady
power output, the rate of release of ionising energy at time t after shutdown at time t ˆ 0 is, from equation (9.2),
dE
P
ˆ
dt
193 MeV†
Z
0
ÿT
2:66
1s
t ÿ t0
1:2
dt 0 MeV sÿ1
" #
1 s 0:2
1 s 0:2
ˆ 0:07P
ÿ
.
t
t‡T
(b) 7.8 MW, 4.3 MW, 0.8 MW.
9.9 The mean free path l of a ®ssion neutron is given by l ˆ 1=nuc , where
nuc is the number density of 239Pu nuclei, and is the total neutron
cross-section. If Rc is the radius of the critical mass M at atmospheric
pressure,
nuc ˆ M=mnuc †= 4=3†R3c :
Chapter 10
259
Hence if Rc ˆ Kl, where K is a constant,
Rc ˆ 4=3†KR3c mnuc =M:
A mass m and its critical radius rc are related by a similar equation, so
that
rc
m 1=2
ˆ
:
Rc
M
If r is the radius of m at atmospheric pressure,
1=3
Rc
M
ˆ
:
m
r
Eliminating Rc ,
r c
r
ˆ
m 1=6
M
:
If m ˆ 0:8 M, rc =r ˆ 0:96.
Chapter 10
10.1(a) 6:5 1014 mÿ2 sÿ1 .
(b) Neutrino mean free path l ˆ 1=nuc and nuc 4 1 A †ÿ3 ˆ 1030 mÿ3 .
Hence l 1015 km 1011 Earth diameters.
10.2 Thermal velocities in the gaseous state of hydrogen exceed the escape
velocity in the Earth's gravitational ®eld. Only hydrogen that is chemically bound remains.
10.3 4.86 MeV.
10.4(a) p ˆ 3:4 1031 mÿ3 ; 2 eÿ ˆ 2:5 10ÿ4 ;
pp ˆ v pp ˆ 1:4 10ÿ49 m3 sÿ1 .
The p±p reaction rate ˆ 12 pp 2p ˆ 8:1 1013 mÿ3 sÿ1 .
Each p±p reaction produces 13.1 MeV and hence the contribution to
" ˆ 170 W mÿ3 .
(b) The p-12C reaction rate ˆ p c pc , and hence the mean time for one
carbon nucleus to react is 1=p pc 106 years.
10.5(a) The reaction rate per unit volume is 2d v and each reaction reduces d
by one. Since this is the dominant reaction (Fig. 10.4)
dd
ˆ ÿ2d v:
dt
(b) This equation can be integrated to give
260
Answers to problems
vt ˆ
1
1
ÿ ;
d 0
and hence the proportion `burned' is
0 ÿ d
0 tc v
:
ˆ
d
1 ‡ 0 tc v
(c) 1020 mÿ3 s:
10.6(a)
dV
e2 =4"0 †
ˆ 0 at r ˆ r0 where r30 ˆ
.
dr
2K
(b) A Taylor expansion of V r† about r ˆ r0 gives
1
V r† V r0 † ‡ M!2 r ÿ r0 †2 ;
2
where M!2 ˆ 3 e2 =4"0 †=r30 .
Classically a particle of mass M in this potential undergoes simple harmonic motion about r ˆ r0 at angular frequency !.
(c) The lowest quantum state in this potential has angular momentum
L ˆ 0 and energy
E0 ˆ V r0 † ‡
1
»!:
2
Taking r0 ˆ 500 fm gives
! ˆ 1:66 1018 sÿ1 ; E0 ˆ 4:87 keV:
(d) Neglecting the Kr2 term in the potential, we may estimate the tunnelling
probability to be exp‰ÿG E0 †Š where
s
e2
2Mc2
G rs =rc †
G E0 † ˆ
E0
»c 4"0
(see equation (6.15)).
Since rs 3 fm and rc 300 fm, G rs =rc † 1 and G E0 † 15:6:
Tunnelling probability ˆ exp‰ÿG E0 †Š ˆ 1:68 10ÿ7 .
(e) In a semi-classical picture, the deuteron and triton approach each other
at intervals of 2=! ˆ 3:8 10ÿ18 s.
The mean number of approaches n before tunnelling (and presumably
fusion) takes place is (cf. }2.3)
n ˆ exp‰G E†Š:
The corresponding time is 2=!†n ˆ 2:3 10ÿ11 s.
Chapter 11
261
Chapter 11
11.1 All states are occupied up to k ˆ kF (equation (B.5)). Using equation
(B.4) the mean value of k is
k ˆ
R kF
0
k3 dk=
R kF
0
k2 dk ˆ 3=4†kF :
For extreme relativistic electrons, E pc ˆ »ck; hence the mean energy
is »ck ˆ 3=4† »ckF .
11.3 The reaction is endothermic and requires an energy input of 0.78 MeV
= Q. The reaction will proceed if "F > Q. (At T ˆ 0 K the proton
Fermi energy is less than the electron Fermi energy EeF by a factor
me =mp .) Using the non-relativistic formula "F ˆ »2 k2F =2me for a
rough estimate, and equation (B.5), the number density of electrons e
must satisfy
ÿ
!32
1 2me c2 Q
e 5 2
ˆ 3 10ÿ9 fmÿ3 :
3
»c†2
The corresponding hydrogen density is 5 109 kg mÿ3 .
11.4 At low temperature, only particles at the top of the Fermi distributions
can take part in the reactions
n ! p ‡ eÿ ‡ e ;
p ‡ eÿ ! n ‡ e :
In thermal equilibrium
"F n† ˆ "F p† ‡ "F e†;
where
"F n† mn c2 ‡
»2
k n†2 ;
2mn F
"F p† mp c2 ‡
»2
k p†2 ;
2mn F
and "F e† »ckF e†; since we expect the electrons to be highly relativistic.
For electrical neutrality, the number density of electrons must equal
the number density of protons. Hence
kF e†3 ˆ kF p†3 ˆ 32 p ;
kF n†3 ˆ 32 n :
The equilibrium condition becomes
mn c2 ‡
»c†2
»c†2
2
2=3
2
‰3
Š
ˆ
m
c
‡
‰32 p Š2=3 ‡ »c‰32 p Š1=3 :
n
p
2mn c2
2mp c2
262
Answers to problems
Taking n ˆ 0:17 nucleons fmÿ3 and solving
p ˆ e ˆ 8:44 10ÿ4 fmÿ3 , and p =n ˆ 5:0 10ÿ3 .
for
p
gives
11.5 If E > E0 ˆ 8 MeV, kB T ˆ 0:5 MeV, then E=kB T > 16. Hence
n
1
2 »c†3
Z
1
E0
E 2 eÿE=kB T dE 1
E02 kB TeÿE0 =kB T ;
2 »c†3
giving n 5 1031 mÿ3 .
11.6(a) From }10.3,
v ˆ
12 32 Z 1
2
m
veÿE=kB T v2 dv
2kB T
0
where E ˆ 12 m =2†v2 , i.e.
1
v ˆ
m kB T
32
1
82 »2 ÿ2
Z
eÿE=kB T
dE:
E ÿ E0 †2 ‡ ÿ2 =4
Integrating over the narrow resonance peak gives the result.
(b) In the plasma, the rate of production of 8Be is 12 2 v (equation (10.6)).
The rate of decay per unit volume is Be = ˆ Be ÿ= ». In equilibrium
these rates are equal.
(c) 2:3 10ÿ10 .
Chapter 12
12.1 fT12 ˆ 4760 s. For a free neutron fT12 ˆ 1015 s, from }12.8. In the simple
shell model the 1s neutron and proton spatial wave-functions would be
the same if Coulomb distortions were neglected, and the spin states
similar to those of a free neutron and free proton. Thus the predicted
fT12 value would be the same. However, since Z ˆ 15, Coulomb distortions are not insigni®cant. Also shell model predictions for the Gamow±
Teller matrix elements, like those for the similar magnetic moment
matrix elements (}5.6), are not accurate.
12.2 Note there is no Coulomb factor in the matrix element.
Ee ˆ 0:71 MeV;
12.3
11
ˆ 3:3 10ÿ20 b:
Be decay: jRf0 j ˆ 0:7 fm,
a nuclear size.
Atomic decay: jRf0 j ˆ 0:4 A , an atomic size.
Chapter 13
263
Chapter 13
13.1 Replacing S0 Ee † in equation (12.5) by E0 ÿ Ee †2 Ee2 , the mean electron
energy is clearly E0 =2 by symmetry. The mean life is inversely proportional to
f Zd ; E 0 † 1
me c2
5 Z
E0
0
E0 ÿ Ee †2 Ee2 dEe ˆ
5
1
E0
:
30 me c2
The proportion of decays within E of the end-point is
Z
1
1 5 E0
1 E02 E 3
E0 ÿ Ee †2 Ee2 dEe :
2
f Zd ; E 0 † me c
f 3 me c 2 †5
E0 ÿE
Substituting for f , the result follows.
13.2 By momentum conservation
p2 c2 ˆ p2Li c2 ˆ 2 6536 MeV 55:9 1:0† eV
ˆ 0:7307 0:0131† MeV2 ;
and by energy conservation
m2 c4 ˆ ‰ 0:862†2 ÿ 0:7307 0:0131Š MeV;
giving 0 m c2 4 160 keV.
13.3(a) From equation (A.4), l ˆ 1= nuc †.
(b) Number of steps = R=l†2 .
Time for each step = l=c.
Total time = R2 = lc†.
13.4 Measuring kB T in MeV,
a ˆ
3 7 2 c
ˆ 1:6 1046 MeVÿ3 sÿ1 mÿ2 :
8 60 »c†3
Since
R1
0
eÿt= dt ˆ , the total energy loss is
a kB T0 †4 4R2 ˆ 3 1058 MeV:
264
Answers to problems
Chapter 14
14.1
Rp T† ˆ mp c2
Z
T=mp c2
0
RM TM † ˆ Mc2 =z2 †
Z
du=F u†:
TM =Mc2
0
ˆ M=z2 mp †mp c2
Z
du=F u†
TM =Mc2
0
du=F u†
ˆ M=z2 mp †Rp mp TM =M†:
p
14.2 T ˆ ‰Mc2 = 1 ÿ v2 =c2 †Š ÿ Mc2
gives
v2
1
ˆ1ÿ
;
c2
1 ‡ u†2
where u ˆ T=Mc2 :
the integration is straightforward.
Taking a constant L,
14.3
14.4
-particle range 20 mm; electron range 1 cm, a much greater distance.
d 1 2
dv
dx dv
dv
ˆM
ˆM :
Mv † ˆ Mv
dx 2
dx
dt dx
dt
From equation (14.4), in the approximation L ˆ L,
dv constant
ˆ
dx
v3
and
dv constant
ˆ
:
dt
v2
Hence
Z
(time to stop)/(range) =
v0
0
v2 dv
.Z
v0
0
v3 dv ˆ 4=3v0 :
For the -particle of question (13.3), time 1:7 10ÿ12 s.
14.5 The kinetic energy of ionising particles is 0.76 MeV. From the end of
}13.1, the number of ion pairs produced is
0:7 MeV=35 eV 2 104 :
The proton will have the longest range. The proton energy is
mt = mp ‡ mt †† 0:76 MeV ˆ 0:57 MeV:
To estimate its range take I ˆ 24 eV (Problem 13.3) and estimate
L 2:5†, which gives a range 0:5 cm.
14.6 From Fig. 13.4, at 50 keV the photon cross-section for lead is predominantly due to absorption and the linear attenuation coef®cient
Chapter 15
265
93:2 cmÿ1 .
For the thickness x to be such that eÿx ˆ 10ÿ3 , we require
x ˆ 0:76 mm.
Chapter 15
15.1 Since the mean life 137Cs is 44 yr, we may neglect its radioactive depletion over 1 yr, and estimate the activity as
137
Cs activity ˆ
0:059 3 109 W† 1 yr†
1:3 1017 Bq:
200 MeV† 44 yr†
If 13% of this activity were spread over 106 km2, the activity per square
metre 2 104 Bq. (Each 235U ®ssion releases 200 MeV energy. See
}9.3.)
15.2 Radon will be produced from the radium decay at a rate R ˆ 1 Ci, but
itself decays. Suppose n t† is the number of radon nuclei after time t.
Then
dn
n
ˆRÿ :
dt
With the initial condition n 0† ˆ 0, the solution of this equation is
n t†
ˆ R 1 ÿ eÿt= † ˆ 1 ÿ eÿt= † Ci:
But n t†= is the radon activity at time t.
After one month, the radon activity will be approximately constant at
R ˆ 1 Ci. Similarly the other decay products up to 210Pb with their even
shorter decay times will be in quasi-equilibrium, each with activity R,
and the total activity will be 6 Ci.
15.3 4 GeV muons are highly relativistic. Hence in equation (14.4) we take
u c. Also Z=A† ˆ 0:5, L ˆ 14, 1 g cmÿ3 ˆ 103 kg mÿ3 . Then
ÿ
dE
ˆ 0:307 0:5 14 MeV cmÿ1 2 MeV cmÿ1 .
dx
Thus a muon passing through the body loses only a small fraction of its
energy to ionisation, and
dE dE dx
dE
ˆ
c
ÿ10ÿ2 J sÿ1 :
dt
dx dt
dx
The number density n of muons in the body is
266
Answers to problems
n
flux
150
mÿ3 5 10ÿ7 mÿ3 :
c
3 108
Hence in 1 s, the received dose is
5 10ÿ7 † 10ÿ2
J 5 10ÿ12 Sv:
103 kg
This suggests an annual dose of 0:16 mSv from this process.
15.4 Taking a total body weight of 70 kg, the activity due to 14C is
3200 Bq. Assuming all the electron energy is deposited in the body
(cf. Table 14.1), the dose per second is
3200 0:052 1:6 10ÿ13 J=70 kg ˆ 3:8 10ÿ13 Sv:
The annual dose is 12 mSv.
15.5 From the given data, a 70 kg body contains 2:47 1020 40K nuclei,
which will yield 1:33 1011 decays per year. If we assume that all an
electron energy, and 10% of a photon energy, is deposited in the body,
the average energy per decay is
Annual dose
‰ 0:89 0:66† ‡ 0:11 0:145†Š MeV
0:6 MeV:
1:33 1011 0:6 1:6 10ÿ13 = 70 kg†
0:18 mSv:
Appendices
B.1(b) The centre-of-mass coordinate R ˆ m1 r1 ‡ m2 r2 †= m1 ‡ m2 †, and
the relative coordinate r ˆ r2 ÿ r1 . The two particle wave-functions
exp ik1 r1 † exp ik2 r2 † and exp ik R† exp ik r† must be identical. The
result follows on equating coef®cients for r1 and r2 . The Jacobian of the
transformation is unity. If the particles are identical, only one hemisphere of the angular integration of the k-vector gives distinct states,
since k and ÿk are equivalent.
C.2 The wave-functions of the state are linear combinations of spatial functions of ®xed l, each of which has parity ÿ1†l . The effect of the parity
operator on the internal states j ‡ 12i, j ÿ 12i of spin 12 fermions (e.g. electrons, protons) is in fact a matter of convention, and they are taken to
have positive parity. It is usually the relative parity of two states which is
signi®cant.
Index
(P) denotes the reference is to a problem
activity, 215
allowed transitions, 168, 174±6
alpha decay, 50±2, 74±82
mean life, 80
series, 82
alpha particles, 33, 40
angular momentum, 16, 230±5
addition of, 233±4
conservation of, 4, 77, 97
intrinsic, 2, 16, 232±3
nuclear, 40, 56, 65
orbital, 230±2
photon, 97
anti-particle, 13, 26
atom, 1
atomic
mass unit, 40
number, 1
attenuation coef®cients, 207±8
axial coupling constant, 176
axial vector, 16
barn, de®nition of, 4
baryon, 29
number, conservation of, 30
becquerel, 215
beta decay, 12, 15, 28, 44±50, 163±79
allowed transitions in, 168, 176
electron capture in, 171
energy spectra, 163, 168±71
Fermi theory, 166±8
mean life, 165, 176
muon, 15
parity violation in, 16, 165, 178
stability conditions, 44±8
stability valley, 48, 49
Bethe formula, 202
`big bang' 151
binding energy, 39±41
of atomic electrons, 39
of last nucleon, 40, see also
separation energy
of light nuclei, 40±1
per nucleon, 48±9
boson, 2
Breit±Wigner formula, 103±6, 107,
110, 241±4
Bremsstrahlung, 205
carbon dating, 55(P)
centre-of-mass system, 223, 230(P)
chain reaction, 119
267
268
Index
Chandrasekhar limit, 154
channel, 99, 223
CNO cycle, 140
compound nucleus, 104
conservation laws
baryon number, 30
charm, 29
electric charge, 15
lepton number, 16
linear and angular momentum, 3
parity, 4, 16
strangeness, 29
symmetry and, 3
cosmic rays, 217
Coulomb barrier
in alpha decay, 76
in nuclear reactions, 109±11, 140
in positron decay, 87(P), 170
critical mass, 121
critical radius, 121, 128(P)
cross-section, 103, 222±7
charged particle, 226
Compton, 209
Coulomb barrier in, 109±11
differential, 34, 224
elastic, 223
®ssion, 116
inelastic, 89, 117, 223
partial, 224
radiative capture, 117, 223
resonant, 103±6, 241±4
Thomson, 211
total, 105, 222
curie, 215
de Broglie relation, 8, 10
decay
channel, 99, 236, 240
rate, 13, 99, 236, 240
rate, partial, 99
delayed neutrons, 86, 118, 122
density of states, 227±30
in beta decay, 169
in gamma decay, 172
integrated, 58, 61, 228
shell model, 58. 61
deuterium, 131, 143
deuteron, 21, 23, 234
quadrupole moment, 23, 25
stripping, 91
dipole, see magnetic dipole moment;
transitions
Dirac equation, 13, 34, 174
direct nuclear reaction, 107, 244
DNA, 214±15
Doppler broadening, 111±12, 124
Einstein mass±energy relation, 8, 10, 39
electric quadrupole moment, 68±72,
73(P)
of deuteron, 23
electromagnetic
®eld, 3, 7
interaction, 8, 12, 179±84
electron, 1, 13
binding energy in atom, 39
capture, 45, 156, 171±3
degeneracy pressure, 152
scattering by nuclei, 34
spin in beta decay, 173±9
electro-weak theory, 9±12, 173
endothermic reaction, 103
energy level diagram, 92
excitation energy, 89
excited states
decay of, 97±9, 236±40
density of, 95
experimental determination, 89±93,
99, 104
mean life, 12, 99
exclusion principle, 2, 43, 58, 152
exothermic reaction, 103
exponential decay law, 12, 236, 238
fermi, de®nition of, 4
Index
Fermi
constant, 175
energy, 58, 72(P), 152
golden rule, 240
interaction, 173
theory of beta decay, 166±8
fermion, 2, 13, 19, 21
Feynman diagram, 9
®ssile nucleus, 116
®ssion, 52
energy release in, 83, 118
induced, 115, 119
mean life, 86, 118
spontaneous, 83±7
¯avour, 29
¯ux, 107, 225
forbidden transitions, 168
f T1/2 value, 185(P)
fusion, 130±5
muon catalysed, 146±8
gamma decay, 97
mean life, 98, 182
theory, 179±83
Gamow±Teller interaction, 173±6
gluon, 3, 22, 26
gravitation, 3, 130, 152±5
gray, 216
hadron, 3, 19±31
half life, 13
hyper®ne structure, 56
impact parameter, 200, 226
internal conversion, 184
ionisation, 199, 206, 209
energy, 203, 213(P)
isobar, 33
isomeric state, 99, 185(P)
isotope, 33, 55(P)
isotopic spin, 22
K-capture, see electron capture
k-space, 227
269
Kurie plot, 188±9
Lawson criterion, 145
lepton, 3, 10, 13±17
liquid drop model, 42, 83, 96
magic number, 62, 65
magnetic dipole moment, 13, 66±8
of nucleon, 20
operator, 67
mass number, 33
mean free path, 119, 225
mean life, 12, 236, see also alpha
decay; beta decay; gamma decay
meson, 26
meson, 26
meson, 27
mirror nuclei, 93, 101(P)
moment, see electric, magnetic
multiple scattering, 206
multi-pole, see transitions
muon, 14, 36
catalysed fusion, 146±8
muonium, 18(P)
neutrino, 14±17, 132, 163
atmospheric, 195±6
burst, 159
cross-sections, 186±8
detector, 142±3, 159, 193
mass, 143, 163, 188±9
mixing, 189±93
oscillations, 192±3
solar, 140±3, 193±5
spin, 16, 188
neutron, 1
cross-sections, 116±18, 222±4
detector, 113(P)
magnetic moment, 20
mass, 19
mean life, 28, 177
radius, 19
star, 154, 157
270
Index
nuclear
angular momentum, 41, 56, 65
binding energy, 39±43
charge distribution, 33±36
electric quadrupole moment, 68±72
®ssion, see ®ssion
fusion, see fusion
magnetic dipole moment, 66±68
magnetic resonance, 67
magneton, 67
mass, 39±43
matter density, 38
parity, 41, 65
potential, 56±8
power, 121±5, 143±6
radius, 39
reaction, 91, 103±13
spin, see angular momentum
time scale, 59
nucleon, 19
±nucleon interaction, 22±8, 52
nucleosynthesis, 155±7, 160±1
pair-production, 14, 208, 211
pairing energy, 40, 43, 65, 116
Paris potential, 24±6, 52
parity, 4, 5, 77, 97, 232, 233
non-conservation of, 16, 165, 178±9
partial decay rate, 99
partial width, 99
Pauli matrices, 233
photodisintegration, 156
photo-electric effect, 209
photon, 3, 8
angular momentum, 97
cross-sections, 207±11, 224
parity, 97
spin, 3, 97
plasma, 132, 135, 144
plutonium, 87(P), 124±5, 129(P)
positron, 13, 45
positronium, 18(P), 30
PP chains, 132, 139±40
prompt neutrons, 85, 118, 123
proton, 1
magnetic moment, 20
mass, 19
radius, 19
pseudo vector, see axial vector
quantum electrodynamics, 14, 210
quark, 3, 21±2, 26±31, 173
rad, 216
radiation
absorbed dose, 216
equivalent dose, 216
man-made sources, 218±19
natural background, 217±18
weighting factor, 216
radiative capture, 105, 116, 160, 223
radon, hazard of, 218
range, 205, 212(P)
reaction rates, 135±9, 225
reactor
control of, 122±4
fast, 122, 125
fusion, 143±6
thermal, 121
thermal stability of, 124
rem, 216
resonant reactions, see Breit±Wigner
formula
risk assessment, 219±20
r-process, 160
Rutherford scattering, 33, 206, 226
scalar potential, 7±8, 10
scattering, see also cross-section
elastic, 223
inelastic, 89, 117, 223
Schmidt values, 68
selection rules
in allowed beta decay, 176
in gamma decay, 182±3
semi-empirical mass formula, 41±3, 83
Index
separation energy, 59, 93, 127(P)
shell model, 41, 48, 56±72, 94
sievert, 216
silicon burning, 156±7
spherical harmonics, 60, 231
spin, see angular momentum
spin±orbit coupling, 63±5
s-process, 160
Standard Model, 31, 175, 192
stopping power, 201±6
strong interaction, 3, 19±28
Sun, 130±3
supernova, 157±60
neutrino burst, 159
symmetry
and conservation laws, 3
energy, 43, 58, 72(P)
tau lepton, 14
tensor potential, 24±5
Thomson scattering, 211, 213(P)
threshold energy, 92
transitions, 97±100, see also decay
electric dipole, 180±2
magnetic dipole, 182±3
multipole, 99, 183
271
tritium, 144, 146, 189
tunnelling, 77±82, 110±11, 155, 170
uranium, 106, 115±27
(1/v) law, 107±8
vector potential, 7,9
virtual process, 9, 15
waste, radioactive, 125
wave-equation, 7, 10, 27
W boson, 3, 9, 15, 28, 167, 174
weak charge, 10
weak interaction, 3, 4, 9±12, 28, 44±8,
131, 134, 163±79
weapons testing, 218
Weinberg angle, 187
white dwarf, 153
width
excited state, 99
partial, 99, 236
resonance, 105, 110±12
total, 236
Z boson, 3, 9, 15